USFWS National Wildlife Refuge Wetland

Ecosystem Service Valuation Model

Phase 1B Final Report

An Assessment of Ecosystem Services Associated with

National Wildlife Refuges

Douglas Patton1

John Bergstrom1

Rebecca Moore2

Alan Covich3

April 2013

Prepared for:

Division of Economics and Division of Refuges

Cooperative Ecosystem Studies Unit (Piedmont)

U.S. Fish and Wildlife Service

Washington, DC

In fulfillment of U.S. Fish and Wildlife Service Award 98210-9-J108 entitled “Valuing

Ecosystem Goods and Service Provided by National Wildlife Refuges”.

The authors would like to thank James Caudill, Erin Carver, and Kevin Kilcullen, all of the U.S.

Fish and Wildlife Service, and the staff of the Division of Economics, U.S. Fish and Wildlife

Service, for their assistance, support, and advice regarding this research and report.

1 University of Georgia - Department of Agricultural and Applied Economics 2 USDI - Bureau of Land Management (former Assistant Professor, University of Georgia) 3 University of Georgia - Odum School of Ecology

1

Executive Summary In recent years, public benefits from ecosystem services have been the focus of increasing

attention from scientists, managers, and stakeholders. The measurement of the non-market benefits such as clean water or flood protection that the public receives from ecosystem services is known as ecosystem service valuation. Primary valuation studies use original data to measure these values and are the ideal approach. Benefit transfer methods have been developed as a low-cost and rapid alternative to primary valuation studies. Benefit transfers are methods where ecosystem service values that have been previously measured by primary valuation studies are used individually or in a group to infer the value of benefits at an unstudied site. Meta-analysis benefit transfer is a robust type of benefit transfer that typically uses a statistical model to incorporate information from multiple primary valuation studies for benefit transfer.

The focus of our research is to develop improved techniques for conducting meta-analysis benefit transfer in order to gain a better understanding of the ecosystem service values associated with wetlands in the USFWS National Wildlife Refuge (NWR) System. In phase 1a of our research we examined how an existing, peer-reviewed meta-analysis model can be used to estimate public benefits from water quality provisioning, flood and storm surge protection, and habitat for commercial fish species that can be attributed to NWR wetlands in four case study NWRs. Phase 1a also included an assessment and quantification of the carbon stocks and associated values of carbon storage at four case study NWRs.

In phase 1b of our research, presented in this document, we conduct a novel meta-analysis restricted to wetlands of the contiguous US, and we expand our analysis of existing meta-analyses by other authors to provide a broad comparison. To illustrate this new approach we focus our novel meta-analysis on the estimation of benefits that wetlands contribute to water quality and flood and storm protection services. We developed a new statistical approach to enhance the accuracy and flexibility of our meta-analysis while maintaining a systematic approach. Using the new modeling method and a new domestic-oriented wetland valuation database, we estimate ecosystem service values at the four case study refuges. Results indicate that wetlands near large populations and with few adjacent wetlands are among the most valuable. Populations with higher incomes are also likely to place a higher value on wetland ecosystem services. Our model also largely agrees with other models that water quality provisioning is often more valuable than flood control and storm protection, yet in many instances (contrary to the other more restrictive models we reviewed) this relationship is found by our flexible model to be reversed. For example, water quality provisioning is found to be less valuable than flood control and storm protection services at the two larger refuges, Blackwater NWR and Okefenokee NWR. Our results broadly indicate decreasing returns to scale as wetland acreage increases, a finding consistent with the comparison models.

Concerns remain in the scientific and public policy communities about the accuracy of benefit transfers. Accordingly, we advise that estimated values based on meta-analysis benefit transfers are valid for making ordinal comparisons across different wetlands or user populations. This approach avoids over-reliance on potentially imprecise values based on small samples and sites with heterogeneous characteristics and user populations. Comparisons of relative differences among estimates and methods will be useful in interpreting the importance of ecosystem service values that often can vary spatially and temporally. Future research is needed to: 1) expand the current model by including a wider variety of primary valuation studies focusing on a broader group of wetland ecosystem services; 2) develop useful applications of these values and test their generality in decision making. For example, these estimated values could be useful along with other information regarding which refuges could be expanded to increase the value of multiple ecosystem services. Ranking the relative public benefits of future wetland acquisitions against the real-estate costs of those acquisitions might be effective in determining landscape-level plans for protecting corridors or providing sustainable habitats under different future demographic or climate scenarios. Novel primary valuation studies of common but under-studied wetlands are also an important means for enhancing our understanding of these important public benefits and enhancing the accuracy of the next iteration of benefit transfers.

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Contents Executive Summary ......................................................................................................................... 1

Introduction ...................................................................................................................................... 3

Methods ........................................................................................................................................... 8

Models from the Literature: Constructing Confidence Intervals and Forecast Combination ...... 8

OLS Regression with a Novel MA Dataset ............................................................................... 11

A Parametric LWLS Regression with a Novel MA Dataset ...................................................... 12

Jackknife Out of Sample Forecast Simulation ........................................................................... 14

Data ................................................................................................................................................ 15

Forecast Combination ................................................................................................................ 15

The Novel MA Dataset .............................................................................................................. 17

Explanatory Variable Values for Forecasting ............................................................................ 22

Results ............................................................................................................................................ 22

Forecast combination ................................................................................................................. 22

OLS ............................................................................................................................................ 24

PLWLS ...................................................................................................................................... 24

Jackknife Leave-One-Out Analysis ........................................................................................... 27

Discussion ...................................................................................................................................... 27

Conclusions .................................................................................................................................... 30

References ...................................................................................................................................... 32

Appendix: A Parametric Locally Weighted Least Squares Regression Procedure ........................ 37

The PLWLS Model Foundation ................................................................................................. 37

Detailed Exposition of the Calibration Procedure ..................................................................... 42

Post Calibration .......................................................................................................................... 45

Figures and Tables ......................................................................................................................... 47

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Introduction This document is intended as a follow-up of the initial assessment of wetland ecosystem services,

“National Wildlife Refuge Wetland Ecosystem Service Valuation Model, Phase 1 Report” by the same

authors. The purpose of this update is to provide additional insights into the process and results of rapid

ecosystem service valuation. The rapid valuation technique we use, Meta-Analysis Benefit Transfer

(MABT), offers researchers a systematic way to test hypotheses about the causes of variations in

ecosystem service values as well as for forecasting ecosystem service values. We are particularly

interested in the potential for using the MABT for producing forecasts of ecosystem service value; this

document is intended in part to provide a fuller picture of the variability of predicted valuation results

from various published meta-analysis studies. We also provide a novel meta-analysis of our own, which is

intended to target water quality provisioning and flood control and storm protection services of wetlands

in continental National Wildlife Refuges. We develop and estimate a novel, parametric locally weighted

least squares regression model that we refer to as PLWLS. The purpose of the new model is to enhance

the efficiency of forecasts.

A wide variety of policy applications ranging from low stakes local decisions to high stakes

international policies require estimates of the benefits to the public provided by various ecosystems. In

order to compare these benefits to the benefits of conventional, private-market goods, quantification is in

terms of monetary value. Benefit transfers (BT) are a valuation approach particularly apt for low-stakes

decision making contexts that tend to exclude quantitative non-market ecosystem service values. Low-

cost, rapid estimates of the economic value of wetland ecosystem services can be produced via benefit

transfer with existing wetland ecosystem service meta-analysis (MA) models, but the accuracy of these

models compared to primary studies is poorly understood. Compounding uncertainty about the

appropriate uses of benefit transfer value estimates obtained from meta-analysis models is the uncertainty

that exists with regard to the correct methodology for calculating benefit transfer estimates. Specifically,

peer-reviewed guidance is lacking with regard to the best practices for implementing a meta-analysis

benefit transfer (MABT), in particular, when several existing meta-analyses are available.

The use of MAs for prediction of a population’s willingness to pay (WTP) for ecosystem services

from wetland landscapes has been, to the best of our knowledge, confined in the peer-review scientific

literature to the very studies that design and estimate an MA model. The process of creating a database of

studies, coded for all relevant explanatory variables and estimating a model requires a considerable

amount of time and expertise. As MABT value estimates are especially low in cost to produce, a natural

extension of the MABT literature is an analysis of the choices faced by an independent analyst with

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limited resources that prevent the estimation of a novel, state-of-the-art MA model. When resources allow

the creation of a targeted MABT model, questions still remain regarding valid applications of the model.

While we interpret the dependent variable in these studies as a measure of Hicksian surplus, or

willingness to pay, important cautions have been raised in the literature about the appropriateness of such

an interpretation of the dependent variable. Welfare measure consistency, discussed in Smith and

Pattanayak (2002) and Bergstrom and Taylor (2006) and again in the review paper of Nelson and

Kennedy (2009), requires (under a strict interpretation) that the studies comprising the meta-data in the

original MA have the same welfare measure. This feature is lacking in all three MAs used for BT herein.

A similar concept, commodity consistency, requires that the services or goods valued in each observation

of the MA also be identical. The need for a greater understanding of ecosystem service values combined

with the extensive heterogeneity in primary valuation studies has led to compromises on the part of MA

researchers who essentially may choose only two, at best, of the following three desirable attributes: a

large dataset, welfare measure consistency, or commodity consistency. Bayesian MA studies that take

advantage of small samples tend to focus on the latter two (Leon-Gonzalez and Scarpa 2008; Moeltner,

Boyle, and Paterson 2007; Moeltner and Woodward 2009), while the studies in our analysis focus on a

large dataset by including a variety of valuation approaches and services. We rely on dummy variables to

achieve commodity and welfare measure consistency.

Meta-analysis methods were originally developed by Glass (1976) for evaluating the effect size

of treatments in the field of education research. The first ecosystem service meta-analysis studies were

published in the late 1980’s and early 1990’s (Walsh, Johnson, and McKean 1989; Smith and Kaoru

1990a; Smith and Kaoru 1990b). A sequence of wetland ecosystem service studies that focused on the

landscape rather than the consumer have been published more recently (i.e., Woodward and Wui 2001;

Brander, Florax, and Vermaat 2006; Ghermandi et al. 2010). These studies are convenient for answering

questions about management of wetland landscapes because the dependent variable is willingness to pay

per acre rather than willingness to pay per person as was the case with earlier nonmarket valuation meta-

analysis studies.

Benefit transfer methodologies such as benefit point transfer preceded MABT by more than a

decade (Johnston and Rosenberger 2010). Benefit point transfers are the simplest type of benefit transfer

where a measure of welfare obtained by studying one site is used as a value for a different site as if it

were measured there by the primary valuation study. Commonly, the site where the original study was

performed is referred to as a study site, while the site where the value is applied is referred to as a policy

site (Rosenberger and Loomis 2000). A variety of techniques have been developed to transfer one or more

study results from study sites to a policy site; while results have been mixed, studies reviewing a variety

of transfer methodologies have often concluded that MABT can lead to lower transfer error than benefit

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transfer methodologies that make less systematic use of the available data, but that existing MA studies

do not meet the necessary criteria and thus may perform quite poorly (Rosenberger and Stanley 2006;

Johnston and Rosenberger 2010). However, MABT models make the most systematic use of available

data, a highly desirable feature in empirical studies.

In this paper, we examine the flows of selected ecosystem services from wetlands in Arrowwood

National Wildlife Refuge (NWR), North Dakota; Blackwater NWR, Maryland; Okefenokee NWR,

Georgia; and Sevilleta and Bosque del Apache NWRs, New Mexico. The Sevilleta and Bosque del

Apache NWRs are modeled as a single unit because of their proximity to one another within a single

ecoregion along the Rio Grande River and the availability of extensive data from the Sevilleta’s Long

Term Ecological Research (LTER) project. The choice of sites is intended to contrast major types of

wetlands of the contiguous United States. Salinity, precipitation, temperature, and distance to ocean are

examples of physical variations across these sites. Variability across the sites in income distribution,

population density, and culture reasonably well represent the range of diversity that could be compared

quickly within the scope of our analysis. All values are adjusted for inflation to 2010 US dollars using the

US Bureau of Labor and Statistics CPI Inflation Calculator. Landcover data for the Brander et al. (2006)

meta analysis primarily come from the NLCD 2006 dataset (Xian et al. 2009) while landcover data for the

Ghermandi et al. (2010) meta analysis come from the USFWS National Wetlands Inventory database

(USFWS 2011), due to the definitions used in these studies to distinguish among wetlands.

Our analysis focuses on two specific wetland-related ecosystem services, water quality

provisioning and flood control and storm protection. In this report we typically refer to the combined

services of flood control and storm protection as simply flood control, as the valuation studies in our

meta-analysis dataset include coastal storm protection services focused on flooding from storm surge.

Similarly, for simplicity, we refer to water quality provisioning simply as water quality. The provisioning

of increased water quality and flood control affect downstream (or inland) populations that are off-site.

Because the pre-existing MA studies we consider do not require an estimate of the population benefitting

from the services and the two services we consider affect populations in a spatially complex manner,

these services can be valued in a much more straightforward manner with a MA that has a dependent

variable normalized by wetland surface area. Recreation benefits, especially in the context of NWRs

which have visitation data, might be more accurately valued using MAs that have the dependent variable

welfare measure normalized by person, trip, or household. Recreation benefits are not modeled in this

study.

Because several relevant meta-analyses of wetland ecosystem services can be found in the peer-

reviewed academic literature (e.g., Woodward and Wui 2001; Brander et al. 2006; Ghermandi et al.

2010), we begin our benefit transfer valuation experiment by exploring the options for predicting

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willingness to pay for a landscape that provides ecosystem services using each of these studies. Due to the

logarithmic transformation of the dependent variable common in ecosystem service meta-analyses, no

simple method exists for computing benefit transfer expected values from stochastic MA models.

Accordingly, to estimate the distribution of willingness to pay, we simulate the dependent variable’s

distribution, allowing for estimation of quantiles and the mean associated with each existing MA and

study site. Because we have three similar MA models pertaining to wetland ecosystem services, we

implement a forecast combination approach in order to efficiently combine data from each of the existing

meta-analyses considered.

The heterogeneity of the primary data for each MA model used in our analysis makes unclear

whether the best interpretation of predicted values is that they are the value of a service or the value of a

wetland that provides a service. In choosing the contingent valuation methodology (CVM) or stated

preference approach, where applicable, the most apt interpretation is that we are valuing a wetland that

provides a service. A review of many of the CVM studies that are referenced by each MA and of the

stated preference studies in our own MA dataset indicates that the valuation questions is typically focused

on a particular wetland extent and the respondents are notified of the most important services provided by

that wetland. The units of the dependent variable in the existing MA models, WTP/area/year, also suggest

that the appropriate interpretation of the MA model dependent variable is associated with a wetland that

provides ecosystem services.

The validity of using existing MA models for MABT depends on the required accuracy of the

intended use of the results (Sargent 2010). An understanding of the features of each study is useful for

developing a qualitative sense of the expected accuracy of each model. The validity of MABT values for

land-use and public policy decisions is still a matter of debate due to uncertainty about the accuracy of

transfers. Much of the focus of this study is not on the validity of BT estimates for policy decisions, but

on the content validity (Bishop 2003) of value estimates obtained with an MA model. In other words, a

primary purpose of this study is to help establish best practices for using existing MAs for MABT.

A key theme that emerges in the steps detailed below is the reduction in accuracy and simplifying

assumptions for our analysis that are required due to unavailability of a thorough statistical description of

each published MA model. Below, we first briefly discuss important features of wetlands in each NWR

study site and our qualitative expectations of the value of several ecosystem services. Next, we discuss the

method followed and data necessary for implementing each MA model and provide a discussion of how

one might best proceed when faced with incomplete modeling information. Then we present the results of

each MABT dependent variable simulation. We end with a discussion of the results and implications for

future MA studies.

7

In addition to analyzing the performance of existing MA studies, we develop a novel MA

database and implement two estimation strategies. The first strategy is the familiar OLS model and the

second strategy is the PLWLS estimator that we develop, primarily in the appendix. The unique MA

database that we develop focuses entirely on wetlands in the contiguous United States. While we are most

interested in flood control and water quality services, we include a variety of other services and necessary

controls in order to obtain better inference on parameters common across all observations, such as the

effect of a population’s income. An important contribution of our MA model is that we normalize the

dependent variable, aggregate willingness to pay, by both acres valued and the population over which the

welfare measure is aggregated. We chose such a normalization and also included the population over

which results are aggregated as an exogenous variable as it became evident in the course of constructing

the MA database that aggregating a welfare measure is too sensitive to potentially arbitrary assumptions

to rely on a reduced form model that omits details about how the analyst chose to aggregate the welfare

measure. Specifically, we found that surface area and population were the most important aspects of the

modeling process that are best treated as analyst imposed decisions for the purpose of answering specific

questions (e.g., What are the benefits of the water quality provisioning services associated with specific

wetlands indicated on a map to the residents of the state that contains the wetland?). An important

implication of this approach is apparent in how we interpret, discuss, and treat the dependent variable.

The alternative interpretation of the MABT model is that we are forecasting the results of a hypothetical

primary valuation study, not that we are directly predicting a welfare measure that can be sensibly

communicated without also describing the methodology by which the value was measured.

The appropriate interpretation of the dependent variable is an important and nuanced issue that in

our view has received too little attention in the ecosystem service MABT literature. In all studies in this

field that we are aware of, the dependent variable is essentially treated directly as an estimate of an

economic welfare concept. We interpret the dependent variable somewhat differently, as the resulting

welfare estimate obtained via a primary valuation study. For forecasts of the dependent variable the

estimated value is fundamentally the predicted result of a primary valuation study. Due to the possibility

of generating predictions from unlikely or even nonsensical combinations of services and methods, such

an interpretation better highlights the context of the WTP estimate and implies the appropriate means for

criterion validation (i.e., conducting a primary valuation study to serve as a comparison). For example, via

MABT one might forecast the value estimated by a travel cost study that values commercial fishing at a

temporarily flooded forested wetland site where no commercial fishing operations exist; criterion

validation via a primary valuation study of this predicted value is not feasible or sensible. Simulations of

the results of such a study with MABT are feasible but the results fail a basic test of content validity.

Similarly, the practice of coding methodological variables at their sample means and predicting a single

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number suggests an unusual (and typically non-linear) hybrid of studies that would defy criterion

validation efforts; alternatively, simulating a number of realistic primary valuation studies (coding

methodological dummy variables as 1’s or 0’s) and using a weighted average to combine the results is

more consistent with the original data and amenable to criterion validation with future primary valuation

studies.

In the sections below, we first develop the basic methodology by which we examine existing MA

studies and the novel database that we have constructed. Secondly we review the data that are required for

the various methodologies we describe. Next we provide the final results and a discussion of those results

followed by some general conclusions. An appendix is included at the end where we develop the PLWLS

estimator in detail.

Methods The methodology section is organized into 4 parts which each describe a means for analyzing

valuation data. These data analyses are intended to both shed light on ecosystem service values estimated

with secondary data and the performance of MA models that are used to analyze and estimate those

values. First we examine the relative sampling performance of existing, published wetland valuation

MAs. Second we present a straightforward OLS model that we will implement with a novel dataset of our

own construction. The third component of our methodology is the introduction of a new estimation

procedure intended to increase the efficiency of MABT forecasts while avoiding ad hoc analyst

resampling. Finally we briefly discuss a means for simulating out-of-sample forecast performance that

can be used to compare the performance of OLS with our new estimator.

Models from the Literature: Constructing Confidence Intervals and Forecast

Combination In our initial analysis, we conduct three separate MABTs, based on the published MA models of

Woodward and Wui (2001), Brander et al. (2006), and Ghermandi et al. (2010). Each of these studies

reports an estimated relationship between per acre (or per hectare) willingness to pay and regressors

related to characteristics of the wetland (e.g., size), the surrounding population and geography (e.g.,

population density), the methodology used to generate the estimate (e.g., contingent valuation, travel

cost), and study quality (only in Woodward and Wui 2001). Table 1 contains a list of explanatory

variables included in each of the three pre-existing meta-analysis models. We divide these variables into

categories similar to the divisions of explanatory variables in each study. All three of these meta-analysis

9

studies include a diverse assortment of primary valuation studies that include several wetland ecosystem

services and represent all populated continents. The first step in our method is a parametric Monte Carlo

simulation of the dependent variables of interest for each meta-analysis. The second step is a forecast

combination procedure that combines the results from the first step for each service at each refuge. We

develop these methods in the context of MABT in order to shed light on questions about how one might

use existing MAs for MABT. We are particularly interested in how one might assess the statistical

properties of MABT forecasts with only the published summary of a model and dataset available.

Monte Carlo Simulation of Distribution As mentioned above, the popularity of using logged dependent variables necessitates a more

complex approach such as bootstrapping or Monte Carlo simulation to capture the effects of curvature of

the exponential function within the expected value operator (Wooldridge 2002). In order to properly

model the exponential function with limited information about the original sample, we simulate the entire

distribution of each dependent variable forecast using a parametric Monte Carlo simulation. The

bootstrapping approach would be desirable, but is not possible without the original sample.

Limited information reported in the three meta-analyses makes it difficult to accurately estimate

confidence intervals associated with the expected WTP estimates without relying on additional

assumptions. Using estimated coefficients and standard errors, we use a Monte Carlo simulation to obtain

a distribution for the dependent variable. The original MA studies we employ do not report the full

covariance matrix of the estimated parameters, so we assume zero covariance between parameters,

resulting in a diagonal variance-covariance matrix for the regression parameters.

The basic algorithm for our parametric Monte Carlo simulation, implemented in Matlab, is to

draw a pseudo-random vector of parameter values from each meta-analysis using a joint normal

distribution with mean and variance equal to the estimated parameters reported in each study. We

multiply each of the constant value regressors from Tables 2 - 5 by the appropriate joint normal randomly

drawn parameter and the sum of these products is exponentiated by the base of the natural logarithm. The

process is iterated one million times for each meta-analysis, each wetland NWR and each of the two

services considered, resulting in 24 vectors with length equal to one million containing log normally

distributed WTP estimates. We compute the mean of each series and the variance of the mean by

regressing each vector on a 1 million element column of 1’s and obtaining the OLS estimated parameters

and associated homoskedastistic variances. Also, from each of the 24 sorted vectors we obtain quantiles

representing the 5th, 50th, and 95th percentiles.

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The parametric Monte Carlo simulation is useful because it provides us with a means for

estimating the variance of each WTP estimate. The popular alternative procedure typically requires an

estimate of the standard error of the residual (Wooldridge 2002), which was not available for all the MAs

we consider. Our assumption that the covariances of the parameters are all zero may lead to substantial

error in the estimated variance of the dependent variable; in particular this may be a problem when used

with a model based on a small sample. For example, if we observe two explanatory variables that have

positive values and we randomly draw positive estimated coefficients that should have negative

covariance but this covariance is assumed to be zero, the resulting sum of the product of each variable and

its coefficient will be biased upwards. The purpose of our experiment is to explore how to implement

MABT using a suite of existing models from published summaries of the original results. Obtaining each

dataset and running full-sample diagnostics would be preferable, but as rapidly growing interest in

ecosystem service values outpaces funding to implement primary valuation studies to estimate these

values, the usefulness of existing models to produce the lowest cost quantitative estimates is a natural

question.

Forecast Combination We use the simulated distribution of WTP described above to combine the three separate point

estimates for each simulated study into a single value via an inverse variance weight approach

(DerSimonian and Laird 1986; Borenstein et al. 2009). The forecast combination literature suggests an

inverse variance weighting approach often leads to improved forecast performance (Smith and Wallis

2009). Accordingly, we take a weighted average of the three median values estimated from each meta-

analysis for each service at each refuge. This inverse variance weighting is identical to implementing an

efficient generalized least squares regression (Wooldridge 2002) where the model includes only an

intercept and we possess known variances. The weighted average of WTP for site i and service j, can be

seen in equation (1), where subscripts i and j index the site and service and subscript k indexes the meta-

analysis used for the WTP estimate. Summation is over k=1,…,3 for the three meta-analyses.

𝑊𝑊𝑊𝑊𝑊𝑊�������𝑖𝑖𝑖𝑖= ∑[ 1

𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚(𝑊𝑊𝑊𝑊𝑊𝑊𝑖𝑖𝑖𝑖𝑖𝑖)]

∑[ 1𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖

] (1)

In equation (1) we choose the point estimate or median of the distribution as a conservative

central estimate of WTP but use the estimated variance of the sample mean of the one million Monte

Carlo draws. The use of a median is consistent with predicting the outcome of a referendum vote, where

the magnitude of any individual’s WTP is relevant to the extent that it tells us how that individual will

vote for a policy. The median also reduces the influence of large outliers relative to the mean.

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Additionally, as the logarithmic transformation of WTP ensures positive values across the simulated

distribution (censoring of non-positive WTP estimates), the use of the median measure of central

tendency opposes to some extent the effect of a truncated predicted distribution(i.e., negative values are

not allowed when the dependent variable is in log form). Ultimately, the use of the median in place of the

mean is particularly compelling with inefficiently estimated models as the mean may be very large due to

outliers. In equation (1), each weight is then the inverse of the simulated variance of the mean associated

with that observation. The sum is normalized by the sum of the inverse variances associated with each

meta-analysis model. Alternatively, inverse variances can be interpreted as precision estimates and each

weight is the share of that observation’s precision among the three MAs.

In addition to the inverse variance or precision weighted average, we also estimate a

conventional, evenly weighted average. We do so in order to highlight the influence of simulated

variances on the final estimate of WTP. We choose the median as our initial estimate of willingness to

pay in the spirit of obtaining conservative estimates. For instances where willingness to accept is the

appropriate construct (e.g., when consumers are deprived of the benefits of an ecosystem service to which

they have a legal right), the larger mean rather than median of each simulated distribution may be more

appropriate.

OLS Regression with a Novel MA Dataset In order to allow comparison between our MA dataset and existing datasets, we provide an OLS

regression that is largely consistent with the existing wetland meta-analyses. Diverging somewhat from

existing wetland MAs, we utilize the dependent variable normalized by both surface area in acres and the

population over which individual welfare estimates were aggregated in the primary valuation study.

Equation (2) is the basic form for our linear MABT forecast model where the dependent variable,

log(𝑊𝑊𝑊𝑊𝑊𝑊𝑖𝑖) is the natural logarithm of the estimated aggregate willingness to pay divided by the

population and acreage counts associated with the aggregate value for observation i.

log(𝑊𝑊𝑊𝑊𝑊𝑊𝑖𝑖) = 𝛽𝛽1 + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠\𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑖𝑖 ∗ 𝛽𝛽2

+ 𝑚𝑚𝑠𝑠𝑠𝑠ℎ𝑜𝑜𝑠𝑠𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑠𝑠𝑜𝑜𝑜𝑜𝑜𝑜𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑖𝑖 ∗ 𝛽𝛽3

+ 𝑜𝑜𝑜𝑜𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐𝑠𝑠𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑖𝑖 ∗ 𝛽𝛽4

+ 𝑠𝑠𝑖𝑖

(2)

In equation (2), study/sitevars is a set of study descriptor variables, including the study area and the

count of the study population over which the value estimate is aggregated; both of these values are used in

the normalization of the dependent variable. We specifically label this group of variables to draw

attention to the fact that the acres and population over which valuation is applied are characteristics of a

12

primary valuation study and often not natural characteristics of a landscape or geographic context.

Methodologicalvars includes dummy variables for the service valued and indicators of whether the

valuation approach is stated preference, revealed preference, or a non-utility theoretic approach such as

replacement cost or avoided damage estimates. We group the remaining variables into the broad category

of contextvars, which includes variables that describe both the user population and landscape in the vicinity

of the primary valuation study site.

We include in the last category of context variables a ratio of local, surrounding wetlands to a

local population count as an indicator of the availability of substitutes outside the study area relative to

the local population. These two variables are measured in acres and a count of people, but are distinct

from the acreage and population counts in the variable group, study/sitevar. Other contextual variables

include the average GDP per capita of the one or more states in close proximity to the study site and

indicators of wetland type. An important feature of our meta-analysis model is the normalization we

chose for the dependent variable. Our choice was motivated by our observation that most primary

valuation studies use effectively obscure reasoning for choosing the population size for aggregating

welfare measures and the boundaries and types of land included in the valuation study. This phenomenon

is most evident in stated preference studies that sample from a particular region (e.g., a state) and ask

participants about a particular, compartmentalized landscape. Studies such as these censor willingness to

pay observations that are outside of that state and thus only estimate aggregate values conditional on the

sampled population. These studies also compartmentalize the landscape by assuming that people’s

preferences or landscape structure and function can be meaningfully geographically isolated. By dividing

the aggregate welfare measure by the count of the modeled population, we control for the arbitrary

influence of censoring WTP of non-sampled populations that have positive WTP. We also control for the

effect of the sampled population by including this count as an explanatory variable. Analogously, we

normalize the aggregate welfare measures by the number of wetland acres valued in the study and include

this acreage count as an explanatory variable.

A Parametric LWLS Regression with a Novel MA Dataset The majority of the technical aspects of our discussion of the locally weighted least squares

estimator are confined to the appendix. The purpose of this section is to lay out an intuitive understanding

of how the parametric locally weighted least squares (PLWLS) approach works. The appendix describes a

two-step process, the first is calibration where an unconventional type of parameter is estimated, and the

second step is the use of the estimated parameter to develop weights for a weighted least squares

regression.

13

The motivation behind developing the PLWLS estimator and going beyond OLS is that

we wish to estimate a regression that is targeted towards each site (i.e., a site of interest or centered site)

for which we desire an estimate of value. The motivation for selecting a potentially unique model for each

site comes from a desire for forecast efficiency, or low prediction error variance. The basic concept

behind the PLWLS regressor is that certain variables, referred to as correspondence attributes, are

indicative of how close observations are to each other. Two observations that are close to each other

provide more information about each other than two observations that are not. We make use of this

correspondence information by applying lower weights to data that are less close to the policy site of

interest. The idea of correspondence has arisen in a number of ecosystem service MA publications, but we

are unaware of any attempts to formalize a systematic procedure for estimating or implementing

correspondence.

An immediate challenge that arises in implementing this idea is that we do not know how

to combine correspondence attributes with very different units. Thus we estimate parameters that serve as

a normalization that allows us to combine dissimilar correspondence attributes. For example if we

suspect that geographic distance between two sites and the difference between two sites in average

income of the local population are both important indicators of correspondence, lacking a quantitative

model, we have only our best judgment to guide us in deciding the relative importance of distance and

difference in income. With PLWLS we are able to estimate parameters, referred to as correspondence

parameters that allow us to add up the correspondence distance between two sites as measured by both

geographic distance and the difference in income. An important challenge is that there is a cost to

applying lower weights to certain observations, as we would be losing useful information if we

incorrectly reduce the weight of a particular observation in the model for a policy site of interest.

Accordingly, we allow correspondence parameters to equal zero (which results in the same OLS

estimator in each regression) or any positive, real number. We also analyze the performance of the

estimator in comparison to OLS in an artificial forecast simulation, described in the next section.

The main divergence of our model from a conventional OLS regression is that we have a

regression for each observation; these regressions are distinguished by potentially unique weights used

during estimation. Equation (3) contains the basic formula for estimating the weight applied to

observation j in the regression tailored to site 𝑠𝑠. In this equation we have the sum of H terms in a negative

exponential function.

𝛀𝛀�𝑖𝑖 𝑖𝑖𝑖𝑖−1 = 𝑠𝑠−∑ |𝑜𝑜ℎ𝑠𝑠−𝑜𝑜ℎ𝑗𝑗|𝛿𝛿�ℎ𝐻𝐻ℎ=1 = 𝑠𝑠−(|𝑜𝑜1𝑠𝑠−𝑜𝑜1𝑗𝑗|𝛿𝛿�1+⋯+|𝑜𝑜𝐻𝐻𝑠𝑠−𝑜𝑜𝐻𝐻𝑗𝑗|𝛿𝛿�𝐻𝐻) (3)

The variable 𝑜𝑜ℎ𝑖𝑖 is a correspondence attribute where the subscript ℎ indicates which attribute (e.g.,

income or geographic location) and the subscript 𝑠𝑠 indicates the observation associated with the attribute.

14

Thus |𝑜𝑜ℎ𝑖𝑖 − 𝑜𝑜ℎ𝑖𝑖| is an unweighted measure of correspondence distance, which is weighted by the

estimated correspondence parameter, 𝛿𝛿ℎ, and the sum of these H terms is the argument of the negative

exponential function. The resulting value, 𝛀𝛀�𝑖𝑖 𝑖𝑖𝑖𝑖−1 , is the jth diagonal element of the diagonal, positive

definite regression weight matrix, 𝛀𝛀𝑖𝑖−1. The desired regression parameter vector, 𝐵𝐵�𝑖𝑖, is calculated

according to the WLS formula in equation (4), where the tilde above a variable indicates that we have

estimated that variable via L-WLS.

𝐵𝐵�𝑖𝑖 = (𝑿𝑿′𝛀𝛀�𝑖𝑖−1𝑿𝑿)−1𝑿𝑿′𝛀𝛀�𝑖𝑖−1𝑌𝑌 (4)

Here 𝑿𝑿 is a (𝑐𝑐 × 𝑘𝑘) matrix of explanatory variables including an intercept term, 𝑌𝑌 is a column vector of

observations of WTP of length n, and 𝐵𝐵�𝑖𝑖 is an n-element column vector of parameter estimates. The

subscript 𝑠𝑠 indicates that the matrix 𝛀𝛀�𝑖𝑖−1 has been estimated for observation 𝑠𝑠, which is the sole source of

information that leads to a potentially unique parameter estimate for site 𝑠𝑠.

Jackknife Out of Sample Forecast Simulation We implement a jackknife or leave-one-out forecast analysis in order to develop a better

understanding of how our model might perform for actual forecasts for policy sites that have no

corresponding valuation study. For the OLS model we drop an observation and estimate the regression

coefficients without that observation, then using those coefficients we forecast the dependent variable and

sample error for each observation. The forecast and sample error of the dropped observation is of

particular interest since that observation is out of sample relative to the estimated parameters and

coefficients. The jackknife provides n2 forecasts that can be used to reconstruct an approximation of the

sampling distribution. Because we lack equations for constructing standard errors that reflect variability in

correspondence coefficients, the jackknife also provides a parallel means for assessing the PLWLS and

OLS model variability.

The jackknife procedure for the PLWLS is very similar, but requires recalibration of the

correspondence coefficients for each omitted observation. This procedure also provides us with a first

approximation of the sampling variability of the correspondence coefficients themselves. A natural

question is whether our PLWLS model outperforms OLS in the jackknife setting.

15

Data

Forecast Combination To conduct the MABT dependent variable simulation, two basic steps are required. First, we must

specify values for the explanatory variables pertinent to each wetland ecosystem and for each published

MA model. Second we treat the estimated regression model from the original study as a stochastic

predictive model with estimated coefficient means and variances which we use to simulate a distribution

of the dependent variable. We use the National Wetlands Inventory geospatial database as the primary

definition of wetlands while the NLCD 2006 dataset provides landscape data for the wetland type

distribution in the Brander et al. (2006) meta-analysis. Refuge boundaries are taken from the USFWS

Cadastral Special Interest layers (USFWS 2009); we specifically use parcels designated as “acquired”

under the variable, “status”.

Population density data are obtained from the SEDAC Gridded Population of the World: Future

Estimates dataset for the year 2010 (CIESIN 2005). GDP data for the state(s) occupied by or immediately

adjacent to each NWR are adjusted for inflation to the appropriate data year for each study using the BLS

CPI Inflation calculator; these data are from the BEA’s SA04 State Income and Employment Summary

dataset (BEA 2012).

While many explanatory variables for MABT can be specified with little difficulty, geospatial

and methodological variables require important assumptions. Because all three MAs estimate non-

constant returns to scale, the geographic extent chosen for the analysis will impact the results. As we are

interested in modeling wetlands in NWRs, we use NWR boundaries as the unit of analysis. Future work is

needed to develop a formal procedure for endogenously identifying the appropriate unit of analysis for

wetland ecosystem service modeling, especially in the context of non-constant returns to scale for wetland

acreage.

Several alternative approaches for the treatment of methodological variables when predicting the

dependent variable can be found in the MABT literature (Moeltner, Boyle, and Paterson 2007; Johnston

and Thomassin 2010). In order to include passive use values in our BT estimates, we simulate a

contingent valuation method for the two earlier meta-analyses that include methodological variables. We

use model B from the Ghermandi et al. (2010) study, which omits methodological variables, as the

study’s authors indicate this is the best model among those estimated. Study quality variables for the

Woodward and Wui (2001) meta-analysis were set to zero, implying that we are predicting the WTP

estimates that would be obtained from a hypothetical, high-quality primary valuation study. Our approach

to coding methodological explanatory variable for dependent variable prediction differs from existing

approaches in the literature in that we view the results as a simulation of the results of a hypothetical

16

primary valuation study. This interpretation of the MABT model suggests that if we wish to estimate a

hybrid of methods, we should not code binary dummy variables as fractions, but rather we could estimate

results as if they were from multiple primary valuation studies and then average these results, perhaps

using sample means for each methodological dummy variable as the weights in the average. We do not

attempt to combine valuation results from multiple primary valuation methodologies in this paper. Rather

we choose first the valuation approach (stated preference approach) most suitable for answering questions

about the total value of ecosystem services, and we choose next the valuation method (contingent

valuation method) most abundant in the MA datasets.

The data used to estimate flood control benefits from each MABT are given in Tables 2 - 5. The

layout of the explanatory variable values matches Table 1. The variables for wetland type must sum to 1,

so for example from Table 2 we can see that Arrowwood NWR is 99.8% fresh marsh (fresh marsh

proportion obtained from NLCD 2006 gridcode 95 – emergent herbaceous wetland) and 0.2% woodland

(woodland proportion obtained from NLCD 2006 gridcode 90 – forested wetland) . The population

density variable in the Brander et al. (2006) meta-analysis is negative because the units employed are the

natural log of 1000people/square kilometer and there are fewer than 1000 people in the average square

kilometer in North Dakota. The many dummy variables that were coded to zero are omitted from these

tables, but can be seen in Table 1. To predict the value of flood control, for example, the flood control

dummy was set equal to 1 and all other service valued dummy variables were set equal to 0. The

analogous coding is used to estimate the value of water quality provisioning.

A general review of the studies referenced in each MA publication indicates that some

observations simultaneously valued more than one service provided by a single wetland site. Specifically,

the methodological variables in Brander et al. (2006) and Ghermandi et al. (2010) are unlike conventional

dummy variables, as they are not perfectly linearly related: for example, one valuation study may use

multiple methods to obtain a result or a result may reflect the value of multiple services. Because no

interaction effects among different services are estimated in any of the models, one resulting restriction is

that the partial effect of predicting WTP for a single service provided by a wetland (e.g., flood control) is

the same as adding that same service to another (e.g., adding flood control to a BT value that also predicts

water quality provisioning value); this restriction can imply negative partial effects for certain services.

For example, with the Brander et al. (2006) MA, valuing flood control alone will always return a positive

WTP estimate, but simultaneously valuing flood control and water quality will have a positive but lower

predicted value, indicating that water quality services have a negative effect. We handle this complication

by always valuing only one service at a time. In the context of our interpretation of the dependent variable

as a simulated study result, we simulate a primary valuation study estimated by a single methodology that

focuses on a single service.

17

The Novel MA Dataset While theory offers little formal guidance for the MA practitioner specifying the model to be

estimated, existing MAs provide some guidance concerning functional form and relevant explanatory

variables. Once the MA practitioner has assembled a list of explanatory variables to include in the model,

obtaining the values of each variable for each observation is the next task. Challenges can potentially

arise when attempting to find measurements of both the dependent variable and the explanatory variables.

In the paragraphs below we describe examples of the process behind measuring the values of variables

and how we address problems that arise. We follow with a discussion on the choice of variables included

in the final model used for MABT. Some variables are included in the database that are not used in the

final model; these variables may be useful for future modeling efforts.

Existing wetland ecosystem service MAs specify the dependent variable in terms of WTP per

acre/hectare or WTP per person. To develop a model that follows the published literature, observations

that are retained in our dataset must share a common normalization or units of the dependent variable –

typically WTP per unit surface area or per person. Careful consideration and additional information is

important when combining primary valuation results with different units and different welfare measures

representing different services obtained from differing valuation approaches. The main assumption of the

meta-analyst is that explanatory variables can be specified that control for these many sources of

variation.

Following similar wetland ecosystem service valuation MA studies, we include a variety of

economic valuation approaches as a component of our econometric identification strategy. By including

multiple primary valuation approaches, we allow for a larger sample with more variation in explanatory

variables. A number of studies were eliminated from consideration due to the author’s reuse of a dataset

associated with an earlier valuation study. Other criteria for eliminating a study from our model include

uncertainty about whether the study focuses on ecosystem services attributable primarily to wetlands or if

the study focuses on a broader landscape including uplands. An important final criterion that determines

both sample size and model specification is the availability of values for all explanatory variables for all

observations. A summary of the final data used in our MA regressions is provided in Table 6. Ultimately,

the dataset we used for analysis contains 82 observations of WTP obtained from 26 primary valuation

studies, and all studies are based on unique datasets. The first column of Table 6 contains variable names,

with the dependent variable (2010 US dollars per 1000 acres per person per year) on the first row below

the header. The second column is the sample mean of the 82 observations and the third column in the

18

table is the variance of the sample mean. All methodological variables are binary or dummy variables as

is the variable, “coastal” for sites that are on the coast of an ocean.

The most frequently used approach for valuing wetland ecosystem services in our dataset is the

stated preference approach. The Contingent Valuation Methodology provides 46 observations and the

remaining 14 are from choice experiments. Both of the stated preference methodologies provide

observations of WTP per person, typically as Hicksian Compensating Variation. Most of the remaining

studies follow the revealed preference approach, all 15 of the revealed preference observations we use are

based on the Travel Cost Methodology. We exclude valuation studies that implement the Hedonic Price

Method because so few of these studies estimate the second stage regression necessary for converting

implicit prices into a demand curve that can be used to estimate consumer surplus. The Hedonic Price

Method also captures private market values that are fundamentally different from the public goods we are

interested in, namely water quality and flood control. The remaining observations retained in our final dataset are from studies that utilize economic

valuation methodologies that are generally considered to be less theoretically appropriate. We lump

together into a single omitted dummy variable studies that utilize damage avoidance and replacement cost

(of capital). We include these studies in order to obtain a variety of observations for flood control and

water quality provisioning services. In order to mitigate concerns that replacement cost studies of water

quality overestimate value, we only use studies where the authors have identified that the Clean Water

Act has led to mandatory water treatment.

Because some primary valuation studies simultaneously value more than one service, we utilize a

variable called joint to allow for us to control for primary valuation studies that value multiple services at

one time. For example, Roberts and Leitch (1997) report value estimates that include habitat, recreation,

and aesthetics in a single stated preference study. For each instance such as this, we code each service

variable to one and code the variable joint to be 3. While a more robust approach would be to include an

interaction term for each combination of services, the approach we employ is useful because it does not

consume excess degrees of freedom and also ensures that we can obtain reasonable identification with a

model that already contains a large number of dummy or binary variables.

In primary valuation studies the measured WTP of a representative agent is often aggregated to

the relevant user population, typically through multiplication (e.g., Bergstrom et al. 1990). More

advanced approaches may segment the population by observable characteristics (e.g., Cooper and Loomis

1993; Bishop et al. 2000) or may include information to estimate distance decay (Sutherland and Walsh

1985). Distance decay is the intuitive concept that people further from a site will benefit less from the site

than people who are closer. However, because not all studies contain estimates of aggregate WTP, this

19

process requires additional information from the MA practitioner. Similar problems arise with the Travel

Cost Method which may report WTP normalized by person, trip, or household.

When additional information is required for estimating aggregate WTP associated with an

observation, the MA practitioner typically must make a decision about how to proceed, potentially

introducing error into the model. Some primary valuation studies may specify the geographic extent from

which the sample was obtained, often delimited by political boundaries (e.g., Cooper and Loomis 1993;

Phaneuf and Herriges 1999), but the study may not report relevant socio-economic data on the population

of the area. For example, the local population count and average income are not reported in many

valuation studies. For instances such as these, we use US Census historical data and BEA (2012) data to

recover population and income information for counties and states associated with the user population.

One of the most important aspects of the data gathering process that may require supplementary

information beyond the primary valuation study is the identification of the geographic extent of the

wetland that is valued. Most existing wetland MAs include a surface area regressor, allowing for non-

constant returns to wetland area, this implies that both aggregate values and values-per-acre may be

misestimated if wetland extent is measured with error. Identifying the extent of the wetland valued in an

observation in the dataset also allows for more complex geospatial analysis of the site’s geographic

context. Thus we imposed the requirement that the study must provide sufficient information to identify

the geographic extent of the studied wetland. We also eliminated from our model studies that provide a

value for extremely large wetland extents, such as the entire country (e.g., Bergstrom and Cordell 1991).

The additional analysis and decisions that the MA practitioner needs to make in order to estimate

the extent of valued wetlands can be usefully illustrated by example. In Breaux, Farber, and Day (1995) a

wastewater treatment wetland in Thibodaux, LA is identified but with too little information in the

publication to identify the site on a map. By searching the Internet for documents relating to the “State

Department of Environmental Quality”, “Thibodaux, Louisiana”, and terms relating to wastewater

treatment and discharge, we found a document describing the site. The site in Twilley and Boustany’s

(1990) report matches closely the description in Breaux, Farber, and Day (1995), including the extent of

treatment wetlands (230 hectares vs. 570 acres, respectively) and both describe a two ridge system with a

drainage canal. Additionally, both reports cite locally relevant studies by William H. Conner and John W.

Day; other similar documents confirming the site were also found. The additional information obtained in

the Twilley and Boustany (1990) document allowed us to find the site in the EPA’s Water Quality

Assessment Status reports (http://watersgeo.epa.gov/mwm/?layer=305B&feature=LA120207_00

&extraLayers=null), providing sufficient information for identifying the boundaries of the site in ArcGis.

We then construct a polygon matching the boundaries of the site, allowing for analysis of the surrounding

population and geography.

20

While the Breaux et al. (1995) study valued a small geographically specific wetland, Bergstrom,

Stoll, Titre, and Wright (1990) value a substantially larger wetland system, estimated at 3.25 million acres

by the authors. Using their figure 3, a map of the counties in the study region, we identify the appropriate

Louisiana counties in ArcGis. Because Bergstrom et al. (1990) attribute all consumer surpluses in their

study to wetlands in the selected counties, we retain all wetlands therein as the scope of valuation. The

procedure for identifying coastal wetlands valued in Costanza, Farber, and Maxwell (1989) was similar.

In their study a section includes estimates of the value of hurricane protection services provided by

Terrebonne Parish coastal wetlands. We define their site first by selecting wetlands in Terrebonne Parish

and then selecting by hand the subset of these wetlands that are between populated areas and the Gulf of

Mexico.

Ultimately, the question of how big the wetland study site associated with a primary valuation

study is can potentially have an unclear answer. Stated preference studies evaluate willingness to pay to

protect or prevent the loss of a precisely defined wetland area (i.e., indicated on a map or with a precise

description) (e.g., Sutherland and Walsh 1985; Sanders, Walsh, and McKean 1991), of a generic and

locally relevant wetland (e.g., Johnson and Linder 1986; Blomquist and Whitehead 1998; Bauer, Cyr, and

Swallow 2004), or of a certain portion of a larger nearby wetland area(e.g., Beran 1995; Bishop et al.

2000; Whitehead et al. 2009; Petrolia and Kim 2011). Even for sites that value wetlands of a well-defined

extent, the wetland ecosystem around or nearby the site may be substantially larger. Attributing the

average per-acre willingness to pay for a smaller subset of wetland acres to all adjacent wetland acres

may lead to a large upward bias in the aggregate welfare measure. Accordingly, when applicable, we

specify the wetland surface area variable as the value (i.e. the stated number of acres valued) discussed in

the primary valuation survey. As we discuss below, the accounting of substitute wetlands in a radius

around the study site as an explanatory variable allows for us to control for changes in WTP that are

related to the broader wetland landscape.

Beyond estimating the size of the wetlands being valued, a further conceptual issue arises when

identifying wetland acreage from a primary valuation study. This occurs when the study values a

hypothetical or generic rather than an actual wetland. Johnston et al. (2002) and Bauer et al. (2004) are

examples of valuation studies where a hypothetical or generic site is valued. Both of these studies,

however, indicate a context that motivated the research; we use this context as the actual site of valuation

in order to be able to obtain information about the geographic context of the wetland.

The relatively recent wetland MAs of Brander et al. (2006) and Ghermandi et al. (2010) include

dummy variables distinguishing wetland type. We follow this approach, but we use continuous variables

representing the proportion of wetlands at the site in each subsystem, as defined by Cowardin et al.

(1979). Because many wetland valuation studies consider wetlands that include a variety of wetland

21

subsystems, this flexible approach can capture more variation in sites and is less prone to measurement

error at the expense of requiring more geospatial analysis. However, because we are using other spatial

variables, there is little difficulty in measuring the wetland subsystem proportions beyond identifying the

boundaries of each site. Ultimately through pre-testing, we eliminate all wetland classification variables

from our regression models; these variables may prove useful in future analyses.

With a well-defined digital map of each wetland, we can assess more complex geospatial

variables and relationships. In order to control for the availability of substitute wetland sites, we follow

Ghermandi et al. (2010) who include a variable for the total area of wetlands in a 50km radius around the

center of each site. However, valuation studies that value geographically extensive wetlands, such as all

the wetlands in a particular state, are likely to be associated with a wider scope for substitution than

studies that value small wetlands. Because of this concern we pursue a more flexible method for

measuring substitutes that varies with wetland surface area. In our model the radius begins at the outer

boundary of the study site (rather than the center of the site) and extends for the number of miles that

equals the cube root of study acres. Thus the wetland substitutes variable for an 8000 acre site would

include all wetland acreage in a 20 mile radius (as the cube root of 8000 is 20) beyond the boundaries of

the site. This functional form was chosen based on our best judgment, and in part due to its positive first

derivative and negative second derivative, which causes the radius to increase with a study’s wetland

acreage but at a decreasing rate. We use the NLCD dataset that is closest in time to the year of data

acquisition to define local substitute wetlands. An important limitation in how we account for substitute

sites is that we do not distinguish among different wetland types. For example, a wetland study site that

consists only of brackish wetlands for which freshwater wetlands may be a poor substitute has included in

the count of substitutes both freshwater wetlands and brackish wetlands.

We use NASA-SEDAC’s Gridded Population of the World v3 dataset

(http://sedac.ciesin.columbia.edu/data/collection/gpw-v3) to count the population around a wetland site in

the same manner as we count substitute wetlands. We include this variable in order to control for

variations in demand for wetland services associated with each site. We included this variable as a

measure of the study and site context, as the local population variable differs from the population over

which a benefit estimate is aggregated. Ultimately, the wetland substitute and local population measures

are used as a ratio variable in our regression equation. The intuition behind this specification is that the

ratio of local people to local wetlands might be indicative of the degree of scarcity of wetlands and

associated ecosystem services.

For the PLWLS estimator, we also include latitude and longitude measured in meters from a

common point in the western hemisphere in the Albers 1983 projection, obtained from analysis in ArcGis

10. Distance is measured in the Euclidean sense; we use the familiar Pythagorean distance formula to

22

calculate the distance between all observations in the sample. This value is used only as a correspondence

attribute, as we are unaware of any straightforward means of accounting for distance between

observations in an OLS regression.

Explanatory Variable Values for Forecasting The application of our MA models for out-of-sample forecasts of NWR ecosystem service

benefits requires little additional data. We use USFWS Cadastral Special Interest Layer polygons with a

status of ‘acquired’ as our definition of the extent of each NWR. These polygons combined with NLCD

2006 land cover data provide us with a means to estimate the surface area of wetlands in each refuge,

which is used to construct the remaining geospatial variables. The population variable that is a study/site

variable (a distinct measure from the local population variable that describes the geographic context of the

site) is set as the mean of study populations in our dataset. We choose this value for simplicity, though the

actual application of results to a policy question would benefit from a more carefully and specifically

defined population count.

Results

Forecast combination The results of our Monte Carlo simulation of WTP per acre can be found in Tables 7 to 12.

Tables 7 to 10 contain estimated quantiles of WTP based on the one million multivariate random normal

parameter draws in our Monte Carlo analysis; each table contains a summary of the simulated distribution

of WTP for flood control and water quality at a refuge. The results for flood control are in the top half and

the results for water quality provisioning are in the bottom half of these tables. Each row of Tables 7 to 10

contains an estimate of WTP per year per acre from a particular meta-analysis regression model. The last

two rows within each service are the precision-weighted and evenly-weighted averages of the three

estimates above. We can see that med(WTP) appears to converge to eE[Xβ]. This is a basic feature of the

lognormal distribution based on a symmetric normal distribution. Tables 11 and 12, respectively, contain

the results of our mean and variance estimates for flood control and water quality. The first numerical

column is our estimate of the mean of the dependent variable after being exponentiated by the

antilogarithm. The second column of numerical data is the simulated variance of the mean ,and each row

contains one of the NWRs evaluated by one of the meta-regressions. A comparison of the dispersion of

WTP estimates from each meta-analysis, indicated clearly both by the variance estimates and the 5th and

23

95th estimated percentiles, suggests that the Ghermandi et al. (2010) meta-analysis is far more precisely

estimated than the two older meta-analyses. While this conclusion might be a result of estimation error

due to our assumption that all sample covariances among parameters were equal to zero, with limited

information the Ghermandi et al. (2010) meta-analysis appears to be the most precise by a substantial

margin. Accordingly, we suggest that the Ghermandi et al. (2010) MA values are the best estimates that

can be obtained from existing MA models and can be used in place of the forecast combination results for

simplicity.

The point estimates in Tables 11 and 12 are most similar between the Brander et al. (2006) and

Ghermandi et al. (2010) models, while the Woodward and Wui (2001) values are substantially larger.

This divergence can be attributed in part to the inclusion in the Woodward and Wui (2001) meta-analysis

of study quality variables, which indicate that lower quality studies systematically produce lower welfare

estimates; as we coded the variables for a high quality study, the estimates are larger than the average

study. Figure 1 is a bar graph containing point estimates of WTP per acre per year for each service at each

refuge and estimated with each MA model. In Figure 1, the left hand axis corresponds to the blue bars for

the Woodward and Wui (2001) point estimates while the right hand axis corresponds to the green and red

bars for the Ghermandi et al. (2010) and Brander et al. (2006) studies. An important feature of the welfare

estimates concerns the differences across MA models in the relative WTP estimates for each refuge.

While both the Brander MA and the Ghermandi MA place Blackwater as having the most valuable

wetlands, The Woodward MA places Blackwater NWR as third most valuable per unit of land. This result

appears to be due largely to the inclusion of socio-economic variables in the later two MA models.

Next we aggregate WTP estimates over all wetland acres in each refuge for each service. Figures

2 and 3 are graphic representations of our estimates of WTP for ecosystem services supported by all NWI

identified wetlands in each refuge. The corresponding numerical values can be found in Table 13 along

with median (obtained via point estimate) WTP and the total count of NWI identified wetlands within

each refuge. As discussed above, by our judgment, the best estimates of WTP are those obtained from the

point estimate method of obtaining the median from the Ghermandi et al. (2010) meta-analysis WTP

distribution, found in the last 8 rows of Table 13. In this case our choice of best is motivated entirely by

the simulated variance of the mean of each forecast. The simulated variances indicate that the Ghermandi

et al. (2010) study produces the most precise estimates and that simulated means are unreliable due to the

assumptions required for estimation.

The results in Table 13 and Figures 1, 2, and 3 generally indicate that for wetland ecosystem

services the Okefenokee NWR is most valuable as a whole with the Blackwater NWR providing

moderately less valuable ecosystem services. The large estimated value of the Okefenokee NWR is

largely due to the extensive wetland surface area, as median values per acre are among the lowest of the

24

four refuges considered across all three MAs. The aggregate values of Arrowwood NWR and Sevilleta &

Bosque del Apache NWR wetland ecosystem services are substantially lower, by a factor of

approximately ten. The primary reason for these lower values is the limited surface area of wetlands in

these two refuges. The values of the average acre of the vast Okefenokee NWR are lowest primarily due

to the decreasing returns to scale relationship estimated in all three MAs. The high values per average

acre of Blackwater NWR wetlands are due largely to the high incomes and population densities coupled

with the refuge’s modest surface area. Generally we find that the results of the MABT with existing MA

studies are in line with our imprecise, qualitative expectations.

OLS The results of our OLS regression can be found below in Table 14. We report Huber-White or

heteroskedasticity robust standard errors, t-statistics, and the p-values associated with the t-statistic for

each parameter. The parameters estimated for both water quality and flood control are significant at the

10% level or better. The water quality parameter is significant at the 5% level; the flood control parameter

just barely misses that mark, suggesting that the parameter estimates are reasonably precise. The

parameter estimate for the number of acres valued is negative and significant at better than the 10% level

with robust standard errors. The parameter for population is similar in magnitude to the parameter for

acres, but the p-value is less than 1%. Because the dependent variable is normalized by acres and

population, the interpretation is that a 1% increase in one of these variables leads to about a 0.5%

decrease in WTP/acre/person, which suggests diminishing returns to expanding the scope of acres valued

or the scope of the population for aggregation. The variable, joint, which indicates how many services

were jointly valued is strongly significant and indicates that adding additional services to a valuation

study results in a reduction of the estimated value relative to a model where the additional services were

valued separately. The results of our OLS forecasts are discussed with the results of the PLWLS estimator

in the next sub-section as well as in the discussion section.

PLWLS As the calibration of correspondence parameters is the first step in the PLWLS estimator, we

present these values first. The results of our PLWLS calibration can be found in Table 15. All

correspondence attributes were standardized by their sample means and variances prior to calibration,

with the exception of Euclidean distance which was converted from meters to 1000’s of kilometers. In

Table 15 we have removed the transformation for correspondence attributes other than distance to

25

facilitate a comparison. The results of the calibration step indicate that GDP and Euclidean distance

between sites are important non-methodological determinants of correspondence among studies in our

sample. Additionally, studies that value similar services and which aggregate over a similar population

have higher correspondence with each other than otherwise. The PLWLS calibration step found that the

number of acres valued by a study was not a determinant of correspondence, a notable result that might be

reassessed in future studies. The moderate values for these parameters and the moderate weights

(typically between 0.01 and 0 .9) that are consequently applied in each regression suggest that the

algorithm is moderately down-weighting observations with poor correspondence, but retaining ample

information for reasonably robust estimation.

The next step in the PLWLS procedure after identifying correspondence parameters is to use

those parameters to calculate a (𝑐𝑐 × 𝑐𝑐) matrix of regression weights for each out-of-sample study/site of

interest. These regression weights are used to calculate a regression for each site and also can be used to

rank observations according to their relative information content. For any given centered site,

observations that are weighted more heavily are assumed to have greater correspondence and therefore be

more informative about the centered site.

We provide several graphs that illustrate correspondence in 2 dimensions; we specifically depict

the NWR study sites as the centered observation in each graph in Figure 4. Figure 4 contains a plot for

each refuge, where each refuge is the centered site indicated by the open circle towards the top-right of

each graph, which is also the graph’s origin. In each graph the horizontal axis represents the weighted

difference between each data point and the centered observation Filled circles that are colored blue

receive lower weights, while magenta circles receive higher weights, indicating greater correspondence.

The four smaller filled circles represent the four case study NWRs, which can be seen to all have the same

population, but varying incomes. The purpose of the figure is to demonstrate that the correspondence

parameters give rise to weights that display a similar pattern across these out-of-sample observations or

policy sites.

While the OLS regression treats each observation equally, the PLWLS approach allows us to rank

observations according to the weight applied in each centered regression. The highest ranked observations

contribute the most information, so the forecasted primary valuation study results are associated most

strongly with the characteristics of the most highly weighted observations. Tables 16-23 contain the ten

observations in each centered regression receiving the highest weights for each of the 4 case study refuges

for both water quality and flood control services. The first four tables are for water quality and the next

four are for flood control and storm surge protection. The top row of each table contains the values for the

centered NWR followed in the second row by the column headings. GDP per capita values are all in 2010

US dollars. Euclidean distances are in kilometers. The regression weights are normalized so that all n=82

26

observations have weights that sum to 82; this normalization facilitates a simple comparison with OLS

weights that are always 1 by assumption. Some rows in these tables appear to be duplicates, which is a

result of variations in site attributes not included in the table such as the number of acres of wetlands

valued. For example, in Table 16 the first 3 observations are from Bishop et al. (2000), which differ in

both the measured value of WTP/person/acre/year and the number of wetland acres valued, which are

both variables not included in these tables.

Generally, one can see that studies that value the same service are always the highest ranked

observations for all 8 examples. High correspondence observations that value the same service as the

centered observation tend to have markedly higher weights than subsequent observations that value other

services; however this is not always the case. In Table 20, for example, the weight applied to the Costanza

et al. (1989) flood/storm surge damage avoidance study is substantially lower than the weights applied to

the higher correspondence flood control studies, which is consistent with the non-zero correspondence

parameters estimated for GDP per capita and Euclidean distance. Both of these variables differ

considerably more between Arrowwood NWR and Costanza et al.’s (1989) Louisiana study site and the

higher ranked prairie pothole observations, as can be seen in the table.

We present the valuation forecasts of a stated preference study of flood control and water quality

benefits for each of our NWR policy sites in Tables 24 and 25. The former table contains the median and

mean of the dependent variable after reversing the log transformation and the values are scaled to 2010

US dollars per thousand acres per thousand people. The latter table contains median and mean aggregate

values where the population is specified as the mean value in our dataset, about 3.5 million people. The

median population is about half as large; the mean population in our dataset is relatively high due to

several observations that aggregate benefits over large populations. The median WTP values are the point

estimates of WTP forecasted by OLS and PLWLS estimators. For each estimator we include a forecast of

both the median and mean values of willingness to pay. The magnitude of the difference between the

median and the mean is inversely proportional to the precision of each model, and the consistently smaller

gap between mean and median for the PLWLS model implies that this model is consistently more precise

than the OLS model. We also provide acreage counts used during estimation and aggregation in Table 25.

Also of note, many of the PLWLS dependent variable estimates, both mean and median, fall between the

mean and median of the OLS estimates, implying that the PLWLS results do not suffer from extensive

bias relative to the results from the unbiased OLS estimator. We highlight the mean value single service

revealed preference study results as we consider these to be the best estimates (i.e., for forecasting the

value of ecosystem services when estimated by a primary valuation study) available from our MA

regression methodology. Under our suggested interpretation of the forecasts of the dependent variable,

27

best estimates include the judgment that a stated preference study of a population with about 3.5 million

people would be the best choice to estimate the total economic value of each service.

Jackknife Leave-One-Out Analysis The results of our jackknife leave-one-out procedure indicate that the additional information used

in the PLWLS forecast may be useful for lowering out-of-sample forecast variances. We found that of 82

observations, the PLWLS algorithm when forecasting an out-of-sample observation produced a lower

variance estimate than OLS 50 times (61%). The leave-one-out forecasts of the PLWLS estimator beat the

full data forecasts of OLS 35 out of 82 times (43%), while the leave-one-out forecasts of the OLS

estimator only beat the full information version of the PLWLS estimator 18 out of 82 times (22%). Future

work may focus on identifying patterns that could be useful for predicting whether OLS or PLWLS is

more likely to produce a more efficient out-of-sample forecast.

Discussion We have combined the values obtained from MABT using 4 wetland meta-analysis datasets in

Tables 26 and 27, which contains results for flood control and water quality provisioning, respectively.

Tables 26 and 27 contain forecasted ecosystem service valuation results from simulated stated preference

studies for each refuge and for the 5 MA models (the last two models are based on our dataset). These

tables include a row of results that are the median value or point estimate. Additionally, the mean values

are included below each row of median values. As mentioned above, an important aspect of evaluating

each model is the magnitude of the model’s error variance. The larger a model’s error variance is, the

larger the divide between the median and mean value will be. The PLWLS model has the smallest gap

between median and mean estimates of WTP, implying that the PLWLS model is the most precise.

The high precision of the PLWLS estimator comes at the expense of introducing variability

through the estimated correspondence parameters; this increased variability is not captured by

conventional, robust standard errors. However, the alternative ad hoc process of choosing a sample that is

thought to have high correspondence with the policy site also introduces variability that is not accounted

for when estimating standard errors. Accordingly with the PLWLS and all other econometric models, it is

important to keep in mind that all results are conditional on the sample and model specification, and in the

case of reported standard errors estimated for the PLWLS regression parameters the values are

conditional on the estimated correspondence coefficients which effectively serve as a means for

resampling the data.

28

For the flood control benefit forecasts using the PLWLS methodology, the top observations

(Tables 20-23) are values from primary studies that use a damage avoidance approach to quantifying

flood control benefits. Pre-testing with a variety of combinations of correspondence attributes indicated

that valuation approach (e.g., stated preference, revealed preference, damage avoidance, or replacement

cost) was generally not a determinant of correspondence for our dataset. Accordingly, forecasting a stated

preference valuation result with a substantial proportion of the information coming from damage

avoidance studies is more appropriate than using a stated preference study that values a different service.

The result that flood control damage avoidance studies and flood control stated preference studies do not

differ in correspondence can also be interpreted as being indicative of low WTP for the passive use value

of flood protection services; in other words, people are unwilling to pay more than the costs of avoided

damages for flood protection services.

All MABT results in this paper can be interpreted as coming from a stated preference study, with

the exception of the estimates obtained from the MA model of Ghermandi et al. 2010, which lacks

controls for study methodology1. While we present these 5 models together in Tables 26 and 27, it is

important to caution that for the existing MA studies only the results of the median calculations have been

made with all necessary information. Another important difference lies in our inclusion of the population

over which benefits are aggregated as a dependent variable. The additional assumptions (that the standard

errors of estimated parameters have zero covariance) required to calculate the mean value of the

dependent variable for the MAs other than our own potentially leads to small-sample inaccuracies. Not

without surprise, we find that the accuracy of mean values obtained from the Monte Carlo assessment of

published MAs is questionable, especially considering how tremendously large some of these estimates

are. Ultimately, we conclude that too much information is lacking to perform adequate benefit transfer

with the information provided in the publication of each published MA study when mean values are of

interest.

We also provide a graphic comparison of the median values obtained from each of the 5 MA

models in Figures 5 and 6, and in these figures we compare the results for the average acre of wetlands.

Both figures have a vertical axis that has been logarithmically transformed to allow us to fit all 5 models

in a single, compact figure. The graphic presentation of the results from all 5 MA models indicates what

we believe is the most reliable strength of using MA models for evaluating ecosystem service values; that

1 This is one reason why the Brander et al. (2006) study was used in the Phase 1a report rather than the

more recent Ghermandi et al. (2010) study as including a precisely estimated (significant) study methodology

variable (for the contingent valuation methodology) allows for more confidence in controlling for welfare measure

consistency.

29

is, MA models appear to be most valid for ranking sites in terms of the relative value of ecosystem

services. For example, in Figure 5 we can see that all but the Woodward and Wui (2001) results indicate

that the average acre of Blackwater NWR supports the greatest (by a substantial margin) value of flood

control services. The same 4 MA models also indicate that the average acre at Arrowwood NWR or

Okefenokee NWR consistently provides flood control services of lower value than the other refuges.

Figure 6 provides a less consistent ranking, relative to the flood control rankings, of water quality

related ecosystem service values among the 4 case study sites. For water quality related ecosystem

services, the wetlands of the Sevilleta and Bosque del Apache are consistently ranked as among the most

valuable and the wetlands of the Okefenokee NWR among the least valuable. However, the relative

ranking of the wetlands of Arrowwood NWR and Blackwater NWR for providing water quality related

services are less consistent across MA models. For both of these figures, it is important to note that for all

models except the PLWLS model the relative ranking across refuges within a model (e.g., in the Brander

et al. (2006) model) cannot change between the two services, a restriction of the single equation OLS

approach when interaction terms are not included.

In comparison to the median per acre values from the Brander et al. (2006) meta-analysis, the

dataset we have compiled indicates dramatically higher median estimates of ecosystem service flows. At

the same time, using the best available data, the mean values obtained from the PLWLS estimator are

generally smaller. Due to the problems with estimating the mean of the dependent variable using the

Brander et al. (2006) model and due to the problematic exclusion of population as an explanatory

variable, we believe the results from our dataset are more conceptually appropriate for MABT.

Additionally, due to the exclusion of non-domestic studies from our dataset and due to the use of the

PLWLS estimator, we expect to have greater correspondence and therefore reduced transfer error with

our model and dataset.

The mean of our PLWLS estimator is what we consider to be our best forecast, conditional on the

population count used during estimation. An important contribution of our paper is the modeling and

associated acknowledgement that the population over which welfare estimates are aggregated in primary

valuation studies is essentially always a choice of the original analyst and dictated neither by the model

chosen by the analyst nor the context of the site associated with the ecosystem services being valued.

Studies that utilize an empirical approach to restricting the population over which benefits are aggregated

are surprisingly rare; Sutherland and Walsh’s (1985) study is the only domestic exception we encountered

in our literature search. Clearly an important next step is to develop a more formal empirical means for

choosing the population over which benefits are aggregated.

Essentially, we follow the existing MA literature in agreeing that methodological covariates can

be difficult to assign for forecasting ecosystem service values when one is simply interested in knowing

30

how an acre of wetlands impacts the welfare of society. We expect that the estimation of a single number

that is free of methodological underpinnings is not the goal of MABT, not anytime soon. Rather, we

reiterate the argument that the best interpretation of the dependent variable obtained from a MA

regression model is a simulated primary valuation study result. As ecosystem service valuation studies are

typically conducted in the context of answering a research or policy question, one must choose a specific

valuation methodology to simulate a primary valuation study. If interest lies in a value that is an average

of results obtained from a variety of methodologies, we recommend simulating each valuation

methodology by coding methodological dummy variables to 1 or 0 and then taking the average (perhaps

with unequal weights) of the final forecasts. We specifically caution against coding dummy variables for

binary concepts as fractions (e.g., coding them at their sample means); if this caution is not followed, the

resulting forecasts will be non-linear functions of multiple valuation methodologies that have no clear

interpretation and no clear means for empirical validation.

The results of the regressions using our dataset produce numbers with a magnitude most

comparable to the values estimated by Woodward and Wui (2001). In comparison to the valuation

forecast results obtained from the median of the Ghermandi et al. (2010) MA model, our OLS values tend

to be about 50 times higher. The mean values simulated for the Ghermandi et al. (2010) study are

substantially closer to the PLWLS mean values obtained with our dataset, though substantial differences

exist. Because we do not have access to information regarding even the average population in the datasets

associated with the 3 previously published MAs, we cannot confidently say that the majority of the

variation in benefits is not due to a difference in the population of beneficiaries that was used to make the

forecast.

Conclusions The process of learning about the behavior of the Locally Weighted Least Squares estimator in

our sample has led us to conclude that the most telling way to understand the estimator is in the context of

penalties. Starting from an OLS regression where no observation is penalized, when we consider a local

regression for site i, observations that are distant (in terms of correspondence) from the central

observation are penalized. The weights are unity less the penalties determined by the parameterized

exponential equation described in the appendix. The process of calibration works by fitting a parametric

variance equation to sample variances and efficient forecasts are made based on patterns identified in the

calibration process. The results of our PLWLS estimator seem promising, yet the impact of clustered

observations and a small sample size on the performance of the model along with a reliable means for

estimating regression coefficient variability due to resampling are still important unanswered questions.

31

The appropriate use of welfare measures is dictated by the questions one wishes to answer. The

relative lack of attention to the difference between mean and median values of the dependent variable

(e.g., Woodward and Wui 2001; Brander, Florax, and Vermaat 2006; Ghermandi et al. 2010; Brander,

Brouwer, and Wagtendonk 2013) is problematic as these two measures of central tendency are

appropriate for answering different questions. Median values may be useful for predicting the outcome of

a vote for which an outcome requires a simple majority. Benefit-cost analysis and in general estimation of

economic benefits on the other hand typically require that one uses the value of the mean of the dependent

variable. For models estimated with the dependent variable in log form, estimation of the median value of

the dependent variable is easy, but this ease comes at the cost of potentially using the wrong value (biased

downwards) which for benefit-cost analysis will systematically lead to biased decisions that lead to a

reduction in social welfare as measured by aggregate benefits due to inadequate protection and restoration

of wetlands that provide important services.

We are ultimately interested in structural methods of modeling the contributions of wetland

ecosystem services to human welfare, and the literature as of yet contains only models that are essentially

reduced form in nature. We use the term structural in the sense that estimated parameters can be

interpreted as causal and in contrast to a reduced form model where causal claims are potentially wrong

due to endogeneity. Smith and Pattanayak (2002) use the term differently, to refer to the origins of the

benefit transfer model in microeconomic theory, which Bergstrom and Taylor (2006) refer to as a strong

structural utility theoretic. Our interpretation of the dependent variable as a valuation study result calls

into question the desire to have a strong structural utility theoretic model derived from an individual

consumer’s assumed utility function. The reason for this divergence is that estimates of a change in

welfare potentially have a very different data generating process with different sources of variability

relative to the underlying changes in welfare that researchers are trying to measure.

Our model is another reduced form model because endogeneity may be present, but by clarifying

the interpretation of the dependent variable and also identifying the population and acreage associated

with a welfare estimate as choices made by the original investigators, we believe that our model has

features closer to a structural model than earlier attempts. Additionally, because we are primarily

interested in using our model for forecasting, efficient forecasts may be more important than the causal

interpretations afforded by structural models. Also relevant to the notion of a structural model is the

question of whether we are forecasting the results of primary valuation studies or directly forecasting

ecosystem service values. The former interpretation, which we recommend, also suggests a different

meaning of causal impacts; the interpretation of a causal partial effect indicated by a regression parameter

is with respect to the results of a primary valuation study. Ultimately, divorcing estimates of WTP from

the methodology that uncovers them is more likely to obscure the nature of ecosystem service value

32

estimates than to provide clarity through a reduction in the contextual details that come along with value

estimates. Concerns that a MABT model can be manipulated to achieve any particular result via changing

methodological explanatory variables can be assessed by examining the appropriateness of the

methodologies coded during MABT for answering the question the values are intended for. Our interest in

the total value of services suggests that stated preference methodologies are the only relevant

methodology to use for forecasting.

Future work will likely increase the efficiency of benefit transfers. Efforts to increase efficiency

can focus on expanding the MA dataset through several avenues: inclusion of non-domestic studies and

appropriate controls, increasing the sample size by adding additional services to the analysis, and

increasing the sample size by more carefully reviewing existing studies and relevant technical reports for

missing information that prohibited their use in the current analysis. Validation and updating of the model

through future primary valuation studies is also an important aspect of increasing the efficiency of

MABT. Work is ongoing to identify common domestic wetland sites that have poor correspondence with

the existing MA dataset, as primary valuation studies of these sites are likely to greatly enhance our

ability to produce efficient forecasts. Increasing the efficiency of MABT models for wetland sites that are

both common and understudied is a particularly promising avenue for future primary valuation studies.

The development of useful applications of MABT models to management decisions may be illustrated

with a comparison of potential refuge acquisitions by comparing a ranked list of real-estate costs with a

ranked list of forecasted welfare changes. Rankings such as these may be useful for finding refuge

acquisitions that are likely to have the greatest benefit to cost ratios.

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37

Appendix: A Parametric Locally Weighted Least Squares

Regression Procedure

The PLWLS Model Foundation Our PLWLS methodology has at its core a GLS-like approach to efficient estimation of

regression coefficients. The GLS method is widely known to produce an estimate of regression

parameters that is more efficient than the conventional OLS estimator when the correct weight matrix is

used (Hayashi 2000, 137). In order to estimate GLS, we require knowledge of the appropriate efficient

weight matrix to use during estimation, which is usually a diagonal matrix where each element is the

respective observation’s known inverse error variance. Typically, however the appropriate weight matrix

is not known and must be estimated. The purpose of the estimator developed below is a new approach to

the estimation of the appropriate weight matrix for any benefit transfer of interest. The process detailed

below broadly consists of an initial calibration stage that estimates the parameters that are used in a

function for estimating the variance of each observation in a sample that is tailored to a specific

observation, followed by a suite of GLS regressions that utilize the estimated parameters from the

calibration stage.

Because we are developing a model based on the correspondence among observations, it is

important to fix the basic idea of our approach before exploring the detailed steps necessary for

estimation. Ultimately, we are interested in estimating an efficient regression for each site in our sample.

Consider a site, 𝑠𝑠, that is the subject of a meta-analysis benefit transfer. The ultimate objective of the

approach we develop is an efficient estimator of Bi which is intended to be used for forecasting benefits

for site 𝑠𝑠. Consider the following meta-regression equation for site 𝑠𝑠.

𝑠𝑠𝑖𝑖 = xj𝐵𝐵𝑖𝑖 + 𝑠𝑠𝑖𝑖𝑖𝑖 (5)

where 𝑠𝑠𝑖𝑖 is a scalar, measured (by a primary valuation study) value or WTP per acre per person, xj is a

row-vector of k explanatory variables for site 𝑗𝑗, 𝐵𝐵𝑖𝑖 is the k element population parameter column-vector

for site 𝑠𝑠, and 𝑠𝑠𝑖𝑖𝑖𝑖 is a scalar, zero mean stochastic population error term associated with the data xj and 𝑠𝑠j

and the parameter vector, Bi. Following the classical regression approach, we wish to estimate the value

of 𝐵𝐵𝑖𝑖 that minimizes the sum of squared errors across the sample of 𝑐𝑐 predictions of the dependent

variable, 𝑠𝑠. 𝐵𝐵𝑖𝑖 is indexed by 𝑠𝑠 to indicate that this parameter is optimized to efficiently forecast the

dependent variable for a particular site, 𝑠𝑠. Estimates of the population parameter, 𝐵𝐵𝑖𝑖, are represented in the

following equation as 𝐵𝐵�𝑖𝑖.

38

y�𝑖𝑖 = xj𝐵𝐵�𝑖𝑖 + �̃�𝑠𝑖𝑖𝑖𝑖 (6)

A tilde over a variable indicates that the variable has been estimated by a regression or similar model. We

are interested in efficient estimates of 𝐵𝐵𝑖𝑖, that is where we expect 𝑠𝑠𝑖𝑖𝑖𝑖2 to be as low as feasible for the

available data. Generally, we are interested in cases where 𝑠𝑠 = 𝑗𝑗 , indicating that the particular choice of 𝐵𝐵

is intended for the data at hand, while cases where 𝑠𝑠 ≠ 𝑗𝑗 are typically not of interest.

The most straightforward and familiar estimator of 𝐵𝐵𝑖𝑖for data 𝑠𝑠𝑖𝑖 and x𝑖𝑖 is the OLS estimator, 𝐵𝐵�𝑖𝑖,

𝐵𝐵�𝑖𝑖 = (𝑿𝑿(𝑖𝑖)′ 𝑿𝑿(𝑖𝑖))−1𝑿𝑿(𝑖𝑖)

′ 𝑌𝑌(𝑖𝑖) (7)

𝑿𝑿(𝑖𝑖) is a (𝑐𝑐𝑖𝑖 × 𝑘𝑘𝑖𝑖) matrix contains ni rows of the ki element data row-vectors xj. Analogously, 𝑌𝑌(𝑖𝑖) is a

(𝑐𝑐𝑖𝑖 × 1) matrix (column vector) of scalar values of 𝑠𝑠𝑖𝑖. The subscript, 𝑗𝑗 may take on values from 1 to ni.

The subscript, (𝑠𝑠), indicates that a particular data matrix is intended for calculating 𝑠𝑠�𝑖𝑖, that is an estimate

of the originally measured value of 𝑠𝑠𝑖𝑖. The typical procedure is to choose or weight by some means the

rows of 𝑿𝑿(𝑖𝑖) and 𝑌𝑌(𝑖𝑖) such that the estimate of 𝑌𝑌𝑖𝑖 is efficient relative to the available information. The

fundamental intuition behind a weighted regression is that certain observations contain less information

about the underlying population from which the sample was drawn, and by correctly reducing the weight

of those less informative observations, we are preserve the relative information content across all

observations. In contrast, if we knew the appropriate weights but we did not use them, the more

informative observations would be effectively downweighted relative to the information they contain

about the population.

It is important to distinguish an efficient estimate of 𝐵𝐵 from the particular 𝐵𝐵𝑖𝑖 that is an efficient

estimator of 𝑌𝑌𝑖𝑖. That is to say we are not necessarily interested in the estimate of 𝐵𝐵 that is most efficient

on average for all observations, but rather for any given observation we are interested in the parameter

that is a priori expected to be most efficient for that observation or a set of observations that share

something in common, such as their vicinity. The distinction between the two reasons for choosing

estimators is most apparent when data are systematically eliminated from consideration for a particular

site 𝑠𝑠 by removing said data from the data matrices used to estimate 𝐵𝐵𝑖𝑖. The motivation for eliminating an

observation is typically that the analyst thinks that observation is relatively uninformative about the site of

interest due, for example, to an analyst’s qualitative expectations of poor study/site correspondence.

In order to quantify the idea of study and site correspondence, we assume that an observation’s

membership in the sub-population centered on site 𝑠𝑠 is a function of observable variables, 𝑜𝑜, referred to

throughout as correspondence attributes. Specifically, when considering the membership of an

observation, 𝑗𝑗, in the population centered about site 𝑠𝑠, we are interested in the measure, 𝑜𝑜(𝑜𝑜𝑖𝑖,𝑜𝑜𝑖𝑖;𝛿𝛿),

where 𝑜𝑜(. ) denotes a weighting function and 𝛿𝛿 is a correspondence parameter that we will estimate.

39

At least two statistical interpretations are available to motivate this procedure. First and foremost, to pair

the above assumptions with the efficient GLS estimator, we can assume that the error variance for

observation 𝑗𝑗 in the population centered on observation 𝑠𝑠 is consistently approximated by the function,

𝑜𝑜(𝑜𝑜𝑖𝑖,𝑜𝑜𝑖𝑖). Second we can alternatively interpret the elements of 𝑜𝑜(𝑜𝑜𝑖𝑖 ,𝑜𝑜𝑖𝑖) as probability weights that

adjust each observation’s weight to be consistent with a sampling frequency such that it is as if the

observation were sampled from the sub-population centered on site 𝑠𝑠. This latter interpretation is

analogous to using weighted least squares to correct a stratified sample. Sites that are closer to the center

of the population centered at site 𝑠𝑠 are more likely to be sampled when sampling is focused on site 𝑠𝑠.

Because we initially start with a sample centered only on the U.S., we use a measure of predicted forecast

error variance fit as the criterion for selecting the function, g. Below we provide more details regarding

the estimation procedure and motivation.

To motivate the interpretation of our approach as a GLS estimator, we must specify the variance-

covariance matrix for the model’s population error term, 𝑠𝑠𝑖𝑖 = 𝑠𝑠𝑖𝑖 − 𝑐𝑐𝑖𝑖𝐵𝐵. A straightforward approach

would be to assume a (𝑐𝑐 × 𝑐𝑐) diagonal error variance matrix,

𝜴𝜴 = 𝑠𝑠𝑠𝑠𝑜𝑜𝑜𝑜[𝑠𝑠12,𝑠𝑠22, … ,𝑠𝑠𝑛𝑛2]. (8)

Where 𝐸𝐸[(𝑠𝑠𝑖𝑖 − 𝑐𝑐𝑖𝑖𝐵𝐵)2] = 𝑠𝑠𝑖𝑖2 is assumed to be the known population variance for the error associated with

observation 𝑠𝑠 from the regression of Y on X and diag[] indicates elements of a diagonal matrix.

Knowledge of this matrix allows us to calculate the efficient GLS estimator of B,

𝐵𝐵𝑔𝑔𝑔𝑔𝑣𝑣 = (𝑿𝑿’ 𝜴𝜴−1 𝑿𝑿)−1 𝑿𝑿’ 𝜴𝜴−1𝑌𝑌. (9)

Our approach diverges from the conventional GLS estimator because we assume that a potentially unique

equation exists for each site 𝑠𝑠, according to equation (5). Thus we define for site 𝑠𝑠, the local variance for

observation 𝑗𝑗, 𝑠𝑠𝑖𝑖𝑖𝑖2 , such that,

𝐸𝐸[(𝑠𝑠𝑖𝑖 − 𝑐𝑐𝑖𝑖𝐵𝐵𝑖𝑖)2] = 𝑠𝑠𝑖𝑖𝑖𝑖2 . (10)

Accordingly, for each site 𝑠𝑠, we can construct a local variance matrix, 𝜴𝜴𝑖𝑖−1,

𝑠𝑠𝑠𝑠𝑜𝑜𝑜𝑜�𝑠𝑠𝑖𝑖12 ,𝑠𝑠𝑖𝑖22 , … ,𝑠𝑠𝑖𝑖𝑛𝑛2 � = 𝜴𝜴𝑖𝑖−1 = 𝜴𝜴𝑖𝑖

−1/2𝜴𝜴𝑖𝑖−1/2, (11)

which under the assumed variance structure allows for the consistent GLS estimate of Bi,

𝐵𝐵𝑔𝑔𝑔𝑔𝑣𝑣𝑖𝑖 = (𝑿𝑿’ 𝜴𝜴𝒊𝒊−1𝑿𝑿)−1 𝑿𝑿’ 𝜴𝜴𝒊𝒊

−1𝑌𝑌. (12)

The alternative interpretation of our estimator is based on the idea that an analyst can correct an

observation that is overrepresented or underrepresented in the data with an inverse probability or

frequency weight. Suppose that we know that an observation 𝑗𝑗 under a random sampling mechanism will

be drawn from the sub-population for site 𝑠𝑠 with probability pij. Suppose we also know that drawing

40

observation 𝑗𝑗 under a random sampling mechanism from the population of all observations is pj. If we

assume the sample of 𝑐𝑐 observations is drawn from the general population of wetland valuation studies,

then we can adjust the frequency of site 𝑗𝑗 in the regression for site 𝑠𝑠 by multiplying the row 𝑗𝑗 of our data

matrices Y and X by �pij pj⁄ � (Dumouchel and Duncan 1983). Accordingly, we treat the correction term

the same as we would the variance, constructing the weight matrix, 𝜴𝜴𝑖𝑖−1,

𝜴𝜴𝑖𝑖−1 = 𝑠𝑠𝑠𝑠𝑜𝑜𝑜𝑜 ��pi1

p1�2

, �pi2p2�2

, … , �pinpn�2�. (13)

The fundamental problem with all WLS estimators is the unobservable nature of efficient

weights. Below we detail an algorithmic approach to estimating weights, allowing for a potentially unique

regression for each observation in the sample, designed for forecasting the dependent variable and

providing an understanding of the validity of the forecast based on the correspondence between the

forecast or policy site and each observation in the sample. The first step in understanding the algorithm

behind our method is an explanation of how we formalize the idea of correspondence as a function of site

attributes, discussed next.

One of the main advantages of estimating a regression centered on each site in the sample is that

by doing so we can allow for variation in parameters as a site’s correspondence attributes change relative

to the sample. The typical regression models used in meta-analysis studies of ecosystem services do not

estimate the wide variety of interaction effects that may exist among variables. The meta-analyses of

wetland ecosystem services discussed above, with the exception of Ghermandi et al. (2010) do not discuss

in much or any depth the possibility that these interaction effects exist. Because we have a complex

sample, restricting the partial effects of regressors to be constant may lead to bias and inefficiency. The

method we develop in this chapter is intended to improve on conventional approaches by allowing any

observation to potentially have unique regression parameters, yet we are not required to estimate a

potentially inefficient model filled with all possible interaction effects.

The process of locally calibrating weights involves the use of differences across cross-sectional

observations. The econometric analysis of time series datasets frequently involves analysis of the

difference in a variable over time; however, cross-sectional studies typically lack this feature because for

cross-section data, no simple analog exists to the chronological ordering of time series variables. While

we lack information indicating an empirical motivation for ordering observations, we can pay closer

attention to the differences among observations in our dataset. The procedure we develop utilizes

information about all 𝑐𝑐2 differences among 𝑐𝑐 observations. For each site, 𝑠𝑠, we can take the absolute

difference between a correspondence attribute at site 𝑠𝑠 (ai) and site 𝑗𝑗 (aj) and arrange them in a column

vector that will have 𝑐𝑐 elements, the 𝑠𝑠th element will be zero. In order to allow for multiple

correspondence attributes we also index attributes by ℎ = 1, … ,𝐻𝐻. Thus for ℎ = 1 we have a1i and a1j.

41

Next, we construct the correspondence attribute difference matrix 𝒅𝒅𝑖𝑖. Equation (14) contains the matrix

representation of 𝒅𝒅𝑖𝑖. Each column represents a paraticular correspondence attribute, and each row

represents the difference between an attribute at observation 𝑠𝑠 and an attribute at another observation in

the dataset.

di=�

|a1𝑖𝑖 − a11| |a2𝑖𝑖 − a21||a1𝑖𝑖 − a12| |a2𝑖𝑖 − a22|

… |a𝐻𝐻𝑖𝑖 − a𝐻𝐻1|… |a𝐻𝐻𝑖𝑖 − a𝐻𝐻2|

⋮ ⋮|a1𝑖𝑖 − a1𝑛𝑛| |a2𝑖𝑖 − a2𝑛𝑛|

⋱ ⋮… |a𝐻𝐻𝑖𝑖 − a𝐻𝐻𝑛𝑛|

�

(14)

A barrier arises when considering how H seemingly disparate attributes might be combined into a

single measure of distance. The primary obstacle overcome by the algorithm below is the identification of

correspondence coefficients that serve as normalizations on each correspondence attribute, which allows

for a single measure of distance. Accordingly, we assume that there exists an (𝐻𝐻 × 1) correspondence

coefficient vector, 𝜹𝜹, composed of H scalar elements, 𝛿𝛿1, … , 𝛿𝛿𝐻𝐻. This vector can be premultiplied by

the (𝑐𝑐 × 𝐻𝐻) attribute difference matrix 𝒅𝒅𝑖𝑖 to generate the argument for the function, 𝑓𝑓(. ) that returns a

vector of weights used in the WLS regression for site 𝑠𝑠. Here we use equation 𝑓𝑓(. ) that takes the

correspondence attribute differences as its argument rather than the similar function, 𝑜𝑜(. ) mentioned

above. Equation (15) demonstrates this procedure for a single site, 𝑠𝑠. The next section details the

procedure for identifying the correspondence coefficient vector, 𝜹𝜹.

𝑓𝑓(𝒅𝒅𝑖𝑖𝜹𝜹) =

𝑓𝑓 � �

|a1𝑠𝑠 − a11| |a2𝑠𝑠 − a21|

|a1𝑠𝑠 − a12| |a2𝑠𝑠 − a22|… |a𝐻𝐻𝑠𝑠 − a𝐻𝐻1|… |a𝐻𝐻𝑠𝑠 − a𝐻𝐻2|

⋮ ⋮|a1𝑠𝑠 − a1𝑐𝑐| |a2𝑠𝑠 − a2𝑐𝑐|

⋱ ⋮

… |a𝐻𝐻𝑠𝑠 − a𝐻𝐻𝑐𝑐|� �

𝛿𝛿1

𝛿𝛿2⋮

𝛿𝛿𝐻𝐻

� �

= 𝑠𝑠𝑠𝑠𝑜𝑜𝑜𝑜(𝛀𝛀𝑖𝑖−1)

(15)

Here, diag(.) is an operator that returns a column vector of diagonal elements of a square matrix. When

diag(.) is applied to a column vector, the result is a sparse matrix with the elements of the column vector

on the diagonal and off diagonal elements are zero. The bottom line of equation (15) contains the WLS

regression weights that are the fundamental source of potential divergence of parameter estimates across

observations.

42

Detailed Exposition of the Calibration Procedure The basic task of the calibration step requires estimation of the correspondence parameter vector,

𝜹𝜹. In order to estimate empirical values for 𝜹𝜹, we employ the Matlab constrained linear optimization

algorithm, fmincon. While a variety of objective functions are potentially available for optimization, we

pursue a GLS approach. While true GLS is not feasible for data with unknown variances, iterative

strategies such as Feasible-GLS are available. The approach detailed below is an alternative to Feasible-

GLS and requires no additional data. The variance of a random variable divided by its own standard

deviation is equal to 1. Consequently, we choose as our objective function to minimize the sum of the

squared distance between sample variances divided by their own estimated variance and 1. An analogous

minimization procedure is to fit the sample variances to the parametric equation that predicts sample

variances; this particular fit is what we detail below. The vector 𝑾𝑾𝑾𝑾𝑾𝑾(𝑖𝑖) containing 𝑊𝑊𝑊𝑊𝑊𝑊(𝑖𝑖)𝑖𝑖 for 𝑗𝑗 =

1, … ,𝑐𝑐𝑖𝑖 serves as the dependent variable in our main weighted regression equation, and we are interested

in the case where 𝑗𝑗 = 𝑠𝑠. Equation (16) contains the basic weighted regression model of interest.

𝛀𝛀𝑖𝑖−½𝑊𝑊𝑊𝑊𝑊𝑊(𝑖𝑖)𝑖𝑖 = 𝛀𝛀𝑖𝑖

−½𝑋𝑋(𝑖𝑖)𝑖𝑖𝐵𝐵𝑖𝑖 + 𝛀𝛀𝑖𝑖−½𝑠𝑠(𝑖𝑖)𝑖𝑖

(16)

The typical objective function is to choose the value for 𝐵𝐵𝑖𝑖 that minimizes the sum of weighted

sample residuals. In our model, we have an earlier step, which requires us to estimate the weights to be

used in the regression for each observation. For each iteration of the first step optimization algorithm,

correspondence parameters conjectures are used to develop the weights used in the regression.

Specifically, we utilize the negative exponential function during calibration to specify the correspondence

weight function, 𝑓𝑓, for the diagonal elements of the weight matrix, 𝛀𝛀𝑖𝑖−1.

𝛀𝛀�𝑖𝑖 𝑖𝑖𝑖𝑖−1 = 𝑠𝑠−∑ |𝑜𝑜ℎ𝑠𝑠−𝑜𝑜ℎ𝑗𝑗|𝛿𝛿�ℎ𝐻𝐻ℎ=1 = 𝑠𝑠−(|𝑜𝑜1𝑠𝑠−𝑜𝑜1𝑗𝑗|𝛿𝛿�1+⋯+|𝑜𝑜𝐻𝐻𝑠𝑠−𝑜𝑜𝐻𝐻𝑗𝑗|𝛿𝛿�𝐻𝐻) (17)

In equation (17), which is equation 𝑜𝑜(. ) mentioned above, the jj subscript on 𝛀𝛀�𝑖𝑖−1 indicates the jth row

and jth column of the inverse of an estimate of the diagonal correspondence weight matrix 𝛀𝛀 for site 𝑠𝑠.

For the final model we estimate, |𝑜𝑜𝐻𝐻𝑖𝑖 − 𝑜𝑜𝐻𝐻𝑖𝑖| is actually estimated by using the Pythagorean Theorem

with latitude and longitude to estimate geographic distance in 1000’s of kilometers. The conjectured

values for the correspondence parameters are used to estimate a weighted regression and residuals. If a

valid model is chosen and the correspondence parameters are correctly chosen, the sample variance of

weighted residuals can be expected to equal 1 on average, alternatively, the sample average of the

residuals is expected to have the same variance as the inverse of the weight applied to that observation.

Sample errors or residuals are found by first using a weighted regression to identify a parameter vector,

43

𝐵𝐵�𝑖𝑖 = (𝑿𝑿(𝑖𝑖)′𝛀𝛀�𝑖𝑖−1𝑿𝑿(𝑖𝑖))−1𝑿𝑿(𝑖𝑖)′𝛀𝛀�𝑖𝑖−1𝑌𝑌(𝑖𝑖) (18)

And using this parameter to produce an (𝑐𝑐𝑖𝑖 × 1) unweighted vector of residuals, which when weighted

will be close to 1 if the appropriate GLS-style weight has been applied in equation (4).

𝑌𝑌(𝑖𝑖) − 𝑿𝑿(𝑖𝑖)𝐵𝐵�𝑖𝑖 = 𝒆𝒆�(𝑖𝑖) (19)

The objective function for the calibration step can then be written as it appears in equation (20). We

further weight the residual from the variance calculation by the weight matrix, 𝛀𝛀�𝑖𝑖−1, as there is less

information in the sample error variance estimates for observations that are less precisely estimated by

regression 𝑠𝑠.

min{𝜹𝜹∈ℝH:𝜹𝜹≥0}

�[(1𝑛𝑛𝑖𝑖 −𝑛𝑛

𝑖𝑖=1

𝛀𝛀�𝑖𝑖−1𝒆𝒆�(𝑖𝑖)2 )′𝛀𝛀�𝑖𝑖−1(1𝑛𝑛𝑖𝑖 − 𝛀𝛀�𝑖𝑖−1𝒆𝒆�(𝑖𝑖)

2 )] (20)

Here 1𝑛𝑛𝑖𝑖 indicates a column vector of 1’s with 𝑐𝑐𝑖𝑖 elements, and the squared term applies to each row of

the vector of residuals.

In order to simplify estimation and to reduce the influence of ad-hoc constraints in our model, we

require that for all values of 𝑠𝑠 = 1, … , n, the data matrices 𝑿𝑿(𝑖𝑖) and 𝑌𝑌(𝑖𝑖) be identical in size, but with rows

potentially weighted. Henceforth the (𝑠𝑠) subscript denotes a data matrix that has row 𝑠𝑠 removed, as we do

not predict the centered observation’s forecast error during calibration to avoid overfitting the model. The

restriction that no observation can be fully dropped from a regression also entails that 𝑐𝑐𝑖𝑖 = 𝑐𝑐 − 1 for all 𝑠𝑠

and that elements of 𝐵𝐵�𝑖𝑖 are associated with the same correspondence attribute for all 𝑠𝑠. With the aid of

substitution, equation (20) can be made more evident. Equation (21) contains an alternative version of the

objective function to be minimized.

44

�[(𝑠𝑠𝑠𝑠𝑜𝑜𝑜𝑜(𝛀𝛀�𝑖𝑖) −𝑛𝑛

𝑖𝑖=1

𝒆𝒆�(𝑖𝑖)2 )′𝛀𝛀�𝑖𝑖−1(𝑠𝑠𝑠𝑠𝑜𝑜𝑜𝑜(𝛀𝛀�𝑖𝑖) − 𝒆𝒆�(𝑖𝑖)

2 )]

= ���𝑠𝑠𝑠𝑠𝑜𝑜𝑜𝑜(𝛀𝛀�𝑖𝑖) − 𝒆𝒆�(𝑖𝑖)2 �′𝛀𝛀�𝑖𝑖−1�𝑠𝑠𝑠𝑠𝑜𝑜𝑜𝑜(𝛀𝛀�𝑖𝑖) − 𝒆𝒆�(𝑖𝑖)

2 ��𝑛𝑛

𝑖𝑖=1

(21)

= ���𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅(𝛀𝛀�𝒊𝒊) − (𝒀𝒀(𝒊𝒊) − 𝑿𝑿(𝒊𝒊)�𝑿𝑿(𝒊𝒊)′ 𝛀𝛀�𝒊𝒊−𝟏𝟏𝑿𝑿(𝒊𝒊))−𝟏𝟏𝑿𝑿(𝒊𝒊)

′ 𝛀𝛀�𝒊𝒊−𝟏𝟏𝒀𝒀(𝒊𝒊)�𝟐𝟐�

′𝛀𝛀� 𝒊𝒊−𝟏𝟏

�𝒅𝒅𝒊𝒊𝒅𝒅𝒅𝒅(𝛀𝛀�𝒊𝒊)𝒏𝒏

𝒊𝒊=𝟏𝟏

− (𝒀𝒀(𝒊𝒊) − 𝑿𝑿(𝒊𝒊)(𝑿𝑿(𝒊𝒊)′ 𝛀𝛀�𝒊𝒊−𝟏𝟏𝑿𝑿(𝒊𝒊))−𝟏𝟏𝑿𝑿(𝒊𝒊)

′ 𝛀𝛀�𝒊𝒊−𝟏𝟏𝒀𝒀(𝒊𝒊))𝟐𝟐��

Here 𝛀𝛀�𝑖𝑖 is a function of the correspondence parameter vector, 𝜹𝜹. The highly non-linear nature in which

the parameters enter the objective function requires numerical optimization. We employ a set of linear

constraints through the fmincon Matlab routine for the correspondence parameters, as expressed in

equation (22). In equation (22), 𝑰𝑰𝐻𝐻 indicates an identity matrix with H diagonal elements, i.e., one for

each correspondence parameter.

[𝑨𝑨𝑚𝑚𝑖𝑖𝑛𝑛][𝛿𝛿] ≤ [𝑏𝑏𝑚𝑚𝑖𝑖𝑛𝑛]

[𝑨𝑨𝑚𝑚𝑣𝑣𝑚𝑚][𝛿𝛿] ≤ [𝑏𝑏𝑚𝑚𝑣𝑣𝑚𝑚]

𝑨𝑨𝑚𝑚𝑖𝑖𝑛𝑛 = −𝑰𝑰𝐻𝐻 , 𝑨𝑨𝑚𝑚𝑣𝑣𝑚𝑚 = 𝑰𝑰𝐻𝐻

𝑏𝑏𝑚𝑚𝑖𝑖𝑛𝑛 = �00⋮0

� , 𝑏𝑏𝑚𝑚𝑣𝑣𝑚𝑚 = �100100⋮

100

�

(22)

Due to the linear restrictions defined in (22), parameter estimates will be neither non-negative nor

large. The former restriction is implemented based on the theoretical notion that an increase in the

distance between the correspondence attributes is expected to not provide additional information about

the centered observation. The latter restriction was implemented to avoid searching for extremely large

parameter values, which was useful for ensuring convergence. When we weight the data by multiplying

correspondence parameters by correspondence attribute differences in the GLS equation, the maximum

weight that can be obtained is 1, which for each regression is applied to the centered observation. We

typically start the parameters in the optimization routine at zero, which results in all 𝑐𝑐 models becoming

conventional, evenly weighted OLS regressions. As a parameter value increases, the impact of an

attribute distance on lowering an observation’s weight also increases, as indicated in equation (23).

45

𝜕𝜕 �𝛀𝛀ijj�

𝜕𝜕(𝛿𝛿ℎ) ≤ 0

𝜕𝜕 �𝛀𝛀ijj�

𝜕𝜕 �𝒅𝒅𝑖𝑖𝑖𝑖ℎ�≤ 0

(23)

Where 𝒅𝒅𝑖𝑖𝑖𝑖ℎ denotes the jth row and hth column of the correspondence attribute distance matrix 𝒅𝒅𝑖𝑖, found

in equation (15). The equality in the first inequality of equation (23) holds when 𝑠𝑠 = 𝑗𝑗 or when 𝑜𝑜𝑖𝑖ℎ = 𝑜𝑜𝑖𝑖ℎ;

this partial derivative implies that as a correspondence parameter increases in magnitude for a given

attribute distance between observations 𝑠𝑠 and 𝑗𝑗, the membership of 𝑗𝑗 in the population centered on 𝑠𝑠 is

reduced or that the variance of observation 𝑗𝑗 in the neighborhood of observation 𝑠𝑠 increases. Similarly, the

second inequality in equation (23) is an equality when 𝛿𝛿ℎ = 0 or when 𝑠𝑠 = 𝑗𝑗.

Post Calibration The entire optimization procedure developed in the preceding section was used entirely for

identifying the correspondence parameters, which apply to all 𝑐𝑐 observations and all of their associated

correspondence distances between the 𝑐𝑐 − 1 remaining observations. From this point the next step is

estimating regression parameters conditional on the correspondence parameters identified during the

calibration step. While the centered regression parameters for all observations are implicitly estimated

during the calibration step, a review of how to obtain these parameters for a given vector of

correspondence attributes is useful for understanding how parameters are generated for out-of-sample

forecasts.

For site 𝑠𝑠, the correspondence attribute distance matrix in equation (15) must be computed, next

using the functional form specified in equation (17), where 𝑓𝑓(. ) = e−(.), correspondence attribute weights

can be computed. Next the regression parameters for the centered site 𝑠𝑠 can be computed by using OLS on

the weighted data or equivalently by using WLS on the unweighted data but with the estimated diagonal

correspondence weight matrix.

The process for generating a regression for an out-of-sample observation is precisely the same as

above, with the exception that that correspondence attribute distance matrix, 𝒅𝒅𝑖𝑖, (and its functional

dependent, 𝛀𝛀�𝑖𝑖) is replaced by it’s out-of-sample counterpart, 𝒅𝒅~𝑖𝑖, where ~𝑠𝑠 indicates that the centered

observation is not in the calibration dataset.

46

𝒅𝒅~𝑖𝑖 = �

|a1~𝑠𝑠 − a11| |a2~𝑠𝑠 − a21|

|a1~𝑠𝑠 − a12| |a2~𝑠𝑠 − a22|… |a𝐻𝐻~𝑠𝑠 − a𝐻𝐻1|… |a𝐻𝐻~𝑠𝑠 − a𝐻𝐻2|

⋮ ⋮|a1~𝑠𝑠 − a1𝑐𝑐| |a2~𝑠𝑠 − a2𝑐𝑐|

⋱ ⋮… |a𝐻𝐻~𝑠𝑠 − a𝐻𝐻𝑐𝑐|

�

(24)

The primary difference between 𝒅𝒅𝑖𝑖 and 𝒅𝒅~𝑖𝑖 is that the latter does not necessarily have zeros in all

columns of the ith row. Analogously, 𝛀𝛀~𝑖𝑖−1 differs from 𝛀𝛀𝑖𝑖

−1 in that the ith diagonal element is not

necessarily equal to 1. The reason for this deviation is straightforward: during calibration the ith row of 𝒅𝒅𝑖𝑖

is always the distance between observation 𝑠𝑠 and itself, which is identically zero; post-calibration the

centered observation is not included in the same manner. Because the attribute weight function is the

negative exponential function, the ith diagonal element of 𝛀𝛀�𝑖𝑖 is 𝑠𝑠−(0∗𝛿𝛿�1+0∗𝛿𝛿�2+⋯+0∗𝛿𝛿�𝐻𝐻) = e−0 = 1. Once

𝛀𝛀�~𝑖𝑖 has been computed, the forecast for the out of sample observation is computed according to the

expression in equation (25).

𝑌𝑌�𝑖𝑖 = 𝑋𝑋𝑖𝑖𝐵𝐵�𝑖𝑖

𝑌𝑌�~𝑖𝑖 = 𝑋𝑋~𝑖𝑖(𝑿𝑿′𝛀𝛀�~𝑖𝑖−1𝑿𝑿)−1𝑿𝑿′𝛀𝛀�~𝑖𝑖

−1𝑌𝑌 (25)

Notably, in equation (25) full data matrices are used to calculate the final regression parameter, 𝐵𝐵�𝑖𝑖.

47

Figures and Tables

*Woodward and Wui 2001 values correspond to left side axis labels and Brander et al. (2006) and Ghermandi et al.

(2010) values correspond to right side axis labels, all values 2010 US dollars per average acre per year

Figure 1: Point Estimates of Annual WTP per Average Acre by NWR Wetlands and Service

$-

$10.00

$20.00

$30.00

$40.00

$50.00

$60.00

$70.00

$-

$200.00

$400.00

$600.00

$800.00

$1,000.00

$1,200.00

$1,400.00

$1,600.00

$1,800.00

Floodcontrol

Waterquality

Floodcontrol

Waterquality

Floodcontrol

Waterquality

Floodcontrol

Waterquality

Arrowwood Blackwater Okefenokee Sevilleta

2010

US

dolla

rs p

er a

cre

per

year

Woodward MA Brander MA Ghermandi MA

48

*Woodward and Wui 2001 values correspond to left side axis labels and Brander et al. (2006) and Ghermandi et al.

(2010) values correspond to right side axis labels, all values 2010 US dollars for all refuge wetlands per year

Figure 2: Arrowwood and Sevilleta & Bosque NWR Point Estimates of WTP for Wetland Ecosystem

Services, Aggregate Annual Values

$0

$20,000

$40,000

$60,000

$80,000

$100,000

$120,000

$140,000

$6,400,000$6,600,000$6,800,000$7,000,000$7,200,000$7,400,000$7,600,000$7,800,000$8,000,000$8,200,000

Flood control Water quality Flood control Water quality

Arrowwood Sevilleta & Bosque

2010

US

dolla

rs p

er re

fuge

per

yea

r

Woodward MA Brander MA Ghermandi MA

49

*Woodward and Wui 2001 values correspond to left side axis labels and Brander et al. (2006) and Ghermandi et al.

(2010) values correspond to right side axis labels, all values 2010 US dollars for all refuge wetlands per year

Figure 3: Blackwater and Okefenokee NWR Point Estimates of WTP for Wetland Ecosystem Services,

Aggregate Annual Median Values

$0

$500,000

$1,000,000

$1,500,000

$2,000,000

$2,500,000

$3,000,000

$3,500,000

$0$20,000,000$40,000,000$60,000,000$80,000,000

$100,000,000$120,000,000$140,000,000$160,000,000

Floodcontrol

Waterquality

Floodcontrol

Waterquality

Blackwater Okefenokee

2010

US

dolla

rs p

er re

fuge

per

yea

r

Woodward MA Brander MA Ghermandi MA

50

Figure 4: Plots of Correspondence Distance for 4 Case Study NWRs for stated preference water quality valuation.

51

Figure 5: MA Forecast Comparison for Flood Control, Annual Median Value per Acre

1

10

100

1,000

10,000

2010

US

dolla

rs p

er a

cre

per y

ear

Arrowwood NWR Floodcontrol

Blackwater NWR Floodcontrol

Okefenokee NWR Floodcontrol

Sevilleta and Bosque delApache NWRs Floodcontrol

52

Figure 6: MA Forecast Comparison for Water Quality, Annual Value per Acre

1

10

100

1,000

10,000

2010

US

dolla

rs p

er a

cre

per y

ear

Arrowwood NWR Waterquality

Blackwater NWR Waterquality

Okefenokee NWR Waterquality

Sevilleta and Bosque delApache NWRs Waterquality

53

Table 1: Variables Used in Three Published Wetland Ecosystem Services Meta-analyses.

Woodward and Wui (2001)

Model C

Brander et al. (2006) Ghermandi et al. (2010) Model B

Land

scap

e

Log (acres) Log (hectares) Absolute value(latitude)

Latitude2

% of wetland identified as: Mangrove

Unvegetated sediment Salt-brackish marsh

Fresh marsh Woodland

Log (hectares) % of wetland identified as:

Estuarine Marine

Riverine Palustrine Lacustrine

Human made dummy Wetland area in 50km radius

Geo

g. &

soci

o-ec

onom

ic Coastal indicator Log (GDP per capita)

Log (population density) Study location indicators:

South America Europe

Asia Africa

Australasia Urban

Ramsar proportion

Medium-low human pressure Medium-high human

pressure High human pressure

GDP per capita Population in 50km radius

Met

hodo

logi

cal

Year of publication Valuation method

indicators: Hedonic

Travel cost Replacement cost Net factor income Producer surplus

Marginal Valuation method indicators:

Hedonic Travel cost

Replacement cost Net factor income

Contingent valuation Production function

Market prices Opportunity cost

Marginal Year of publication

Serv

ice

valu

ed

Flood control Groundwater recharge

Water quality Commercial fishing Recreational fishing

Amenity Erosion reduction

Bird watching Non-use appreciation of

species

Flood control Water supply Water quality

Commercial fishing and hunting

Recreational fishing Recreational hunting

Amenity Fuel wood

Biodiversity Material

Flood control Surface/groundwater supply

Water quality Comm. fishing and hunting

Recreational fishing Recreational hunting

Amenity and aesthetics Fuel wood

Natural habitat, biodiversity Harvesting of natural

materials Nonconsumptive recreation

54

Stud

y qu

ality

Published results Poor data quality

Poor theory Poor econometrics

Table 2: Variable Values for Arrowwood NWR Flood Control MABT with Each Published Meta-

analysis.

Arrowwood NWR - Flood Control Woodward and Wui (2001)

Model C Brander et al. (2006) Ghermandi et al. (2010)

Model B Variable Value Variable Value Variable Value Intercept = 1 Intercept = 1 Intercept = 1

Land

scap

e Log acres = 8.43 Log hectares = 7.53 Log hectares = 7.53

- - Abs. val. Lat. = 47.24 - -

- - Latitude2 = 2231 - -

Wet

land

ty

pe (r

atio

s)

- - Fresh marsh Woodland All others

= 0.998 = 0.002

= 0

Palustrine Lacustrine Riverine

All others

= 0.45 = 0.55 = 0.003

= 0

Soci

o-ec

onom

ic

- - Log (GDP per capita) = 10.41 Log (GDP per

capita) = 10.59

- - Log (Pop. Den.) = -5.89 Log(Pop. in 50km) = 9.99

- - Ramsar Proportion = 0 Human pressure

indicators = 0

Geo

g.

indi

cato

rs

Coastal = 0 All = 0 - -

Met

hod-

olog

ical

- - Marginal = 0 Marginal = 0

Year of publication = 52 - - Year of publication = 36

Val

uatio

n m

etho

d in

dica

tors

All = 0 CV All others

= 1 =0 - -

Serv

ices

va

lued

in

dica

tors

Flood control All others

= 1 = 0

Flood control All others

= 1 = 0

Flood control All others

= 1 = 0

55

Stud

y qu

ality

in

dica

tors

All = 0 - - - -

56

Table 3: Variable Values for Blackwater NWR Flood Control MABT with Each Published Meta-analysis.

Blackwater NWR - Flood Control Woodward and Wui

(2001) Model C Brander et al. (2006) Ghermandi et al. (2010)

Model B Variable Value Variable Value Variable Value Intercept = 1 Intercept = 1 Intercept = 1

Land

scap

e Log acres = 10.11 Log hectares = 9.20 Log hectares = 9.20

- - Abs. val. Lat. = 38.4 - -

- - Latitude2 = 1474.6 - -

Wet

land

type

(r

atio

s)

- -

Fresh marsh Woodland

Salt-brackish marsh

All others

= 0.01 = 0.40 = 0.59

= 0

Estuarine Palustrine Lacustrine Riverine

= 0.23 = 0.27 = 0.003 = .0006

Soci

o-ec

onom

ic

- - Log (GDP per capita) = 10.55 Log (GDP per

capita) = 10.73

- - Log (Pop. Den.) = -3.47 Log(Pop. in 50km) = 12.65

- - Ramsar Proportion = 1 Human pressure

indicators = 0

Geo

g.

indi

cato

rs

Coastal = 0 All = 0 - -

Met

hod-

olog

ical

- - Marginal = 0 Marginal = 0

Year of publication = 52 - - Year of publication = 36

Val

uatio

n m

etho

d in

dica

tors

All = 0 CV All others

= 1 =0 - -

Serv

ices

va

lued

in

dica

tors

Flood control

All others

= 1 = 0

Flood control All others

= 1 = 0

Flood control All others

= 1 = 0

Stud

y qu

ality

in

dica

tors

All = 0 - - - -

57

Table 4: Variable Values for Okefenokee NWR Flood Control MABT with Each Published Meta-

analysis.

Okefenokee NWR - Flood Control Woodward and Wui (2001)

Model C Brander et al. (2006) Ghermandi et al. (2010)

Model B Variable Value Variable Value Variable Value Intercept = 1 Intercept = 1 Intercept = 1

Land

scap

e Log acres = 12.84 Log hectares = 11.9 Log hectares = 11.9

- - Abs. val. Lat. = 30.84 - -

- - Latitude2 = 950.92 - -

Wet

land

ty

pe (r

atio

s)

- - Fresh marsh Woodland All others

= 0.12 = 0.88

= 0

Palustrine Lacustrine Riverine

All others

= 0.976 = 0.024 = .0004

= 0

Soci

o-ec

onom

ic

- - Log (GDP per capita) = 10.27 Log (GDP per

capita) = 10.45

- - Log (Pop. Den.) = -5.60 Log(Pop. in 50km) = 10.95

- - Ramsar Proportion = 1 Human pressure

indicators = 0

Geo

g.

indi

cato

rs

Coastal = 0 All = 0 - -

Met

hod-

olog

ical

- - Marginal = 0 Marginal = 0

Year of publication = 52 - - Year of publication = 36

Val

uatio

n m

etho

d in

dica

tors

All = 0 CV All others

= 1 =0 - -

Serv

ices

va

lued

in

dica

tors

Flood control All others

= 1 = 0

Flood control All others

= 1 = 0

Flood control All others

= 1 = 0

Stud

y qu

ality

in

dica

tors

All = 0 - - - -

58

Table 5: Variable Values for Sevilleta and Bosque del Apache NWRs Flood Control MABT with Each

Published Meta-analysis.

Sevilleta & Bosque del Apache NWRs - Flood Control Woodward and Wui (2001)

Model C Brander et al. (2006) Ghermandi et al. (2010)

Model B Variable Value Variable Value Variable Value Intercept = 1 Intercept = 1 Intercept = 1

Land

scap

e Log acres = 8.54 Log hectares = 7.63 Log hectares = 7.63

- - Abs. val. Lat. = 34.05 - -

- - Latitude2 = 1159.6 - -

Wet

land

ty

pe (r

atio

s)

- - Fresh marsh Woodland All others

= 0.79 = 0.21

= 0

Palustrine Lacustrine Riverine

All others

= 0.575 = 0.062 = 0.363

= 0

Soci

o-ec

onom

ic

- - Log (GDP per capita) = 10.24 Log (GDP per

capita) = 10.42

- - Log (Pop. Den.) = -5.45 Log(Pop. in 50km) = 10.24

- - Ramsar Proportion = 0 Human pressure

indicators = 0

Geo

g.

indi

cato

rs

Coastal = 0 All = 0 - -

Met

hod-

olog

ical

- - Marginal = 0 Marginal = 0

Year of publication = 52 - - Year of publication = 36

Val

uatio

n m

etho

d in

dica

tors

All = 0 CV All others

= 1 =0 - -

Serv

ices

va

lued

in

dica

tors

Flood control All others

= 1 = 0

Flood control All others

= 1 = 0

Flood control All others

= 1 = 0

Stud

y qu

ality

in

dica

tors

All = 0 - - - -

59

Table 6 MA Data Summary

Variable Mean Variance n=82

Dependent Value/1000ac/pop† 24.29416996 13584.72966

Study/Site

Acres valued† 154934.3668 2.29501E+11 Population† 3531250.744 5.66455E+13

Methodological (1/0) Valuation Approach Revealed preference 0.18293 0.15131

Stated preference 0.73171 0.19874 Joint valuation (1,2,3) 1.1098 0.17299

Service Valued Water quality 0.12195 0.1084

Flood protection 0.060976 0.057964 Total value 0.5 0.25309

Recreation, general 0.12195 0.1084 Habitat 0.085366 0.079042

Recreation, fishing 0.02439 0.024089 Recreation, hunting 0.15854 0.13505

Interaction_use 0.073171 0.068654 Interaction_passive 0.20732 0.16637

Context

GDP(state)† 31205.5105 37717998.95 Coastal (1/0) 0.12195 0.1084

GDP*local_pop† 25825016098 2.01947E+21 Local_pop:local_wet† 47.06399473 0.057448

† mean and variance in levels, but variable used in log form

60

Table 7: Arrowwood NWR MABT Monte Carlo Quantiles

Arrowwood NWR Predicted per-acre value,

eE[Xβ]

Simulated distribution of expected per-acre value

5th percentile 50th percentile 95th percentile

Floo

d C

ontro

l

Woodward and Wui 1532.83 4.56 1528.94 513829.36

Brander et al. 2.65 0.0000005 2.61 14011026.33

Ghermandi et al. 17.48 0.19 17.56 1634.44 Precision weighted

average 17.49

Unweighted average 517.66

Wat

er Q

ualit

y

Woodward and Wui 1625.99 4.89 1621.53 544059.38

Brander et al. 4.33 0.0000008 4.28 23459222.04

Ghermandi et al. 23.48 0.25 23.58 2176.20 Precision weighted

average 23.49

Unweighted average 551.27

61

Table 8: Blackwater NWR MABT Monte Carlo Quantiles

Blackwater NWR

Predicted per-acre value,

eE[Xβ]

Simulated distribution of expected per-acre value

5th percentile 50th percentile 95th percentile

Floo

d C

ontro

l

Woodward and Wui 844.86 1.90 843.77 370704.19

Brander et al. 13.91 0.0000057 13.75 34432273.25

Ghermandi et al. 46.16 0.42 46.43 5155.70 Precision weighted

average 46.17

Unweighted average 301.64

Wat

er Q

ualit

y

Woodward and Wui 896.22 2.05 893.67 392006.84

Brander et al. 22.70 0.0000090 22.38 58489215.98

Ghermandi et al. 62.00 0.56 62.36 6918.91 Precision weighted

average 62.03

Unweighted average 326.97

62

Table 9: Okefenokee NWR MABT Monte Carlo Quantiles

Okefenokee NWR Predicted per-acre value,

eE[Xβ]

Simulated distribution of expected per-acre value

5th percentile 50th percentile 95th percentile

Floo

d C

ontro

l

Woodward and Wui 435.00 0.89 433.94 209566.46

Brander et al. 4.96 0.0000037 4.91 6702403.62

Ghermandi et al. 5.27 0.05 5.29 550.86 Precision weighted

average 5.27

Unweighted average 148.41

Wat

er Q

ualit

y

Woodward and Wui 461.42 0.97 460.82 221840.58

Brander et al. 8.09 0.0000058 8.01 11151922.26

Ghermandi et al. 7.07 0.07 7.11 741.71 Precision weighted

average 7.07

Unweighted average 158.86

63

Table 10: Sevilleta & Bosque del Apache NWRs MABT Monte Carlo Quantiles

Sevilleta & Bosque del Apache NWRs

Predicted per-acre value,

eE[Xβ]

Simulated distribution of expected per-acre value

5th percentile 50th percentile 95th percentile

Floo

d C

ontro

l

Woodward and Wui 1487.34 4.39 1483.62 501931.60

Brander et al. 6.07 0.0000041 6.00 8991305.62

Ghermandi et al. 18.44 0.20 18.51 1666.25 Precision weighted

average 18.44

Unweighted average 503.95

Wat

er Q

ualit

y

Woodward and Wui 1577.75 4.71 1572.97 531797.40

Brander et al. 9.91 0.0000065 9.77 15247340.97

Ghermandi et al. 24.76 0.28 24.89 2227.39 Precision weighted

average 24.76

Unweighted average 537.47

64

Table 11: Flood Control Monte Carlo Mean and Variance

Flood control

dollars per acre per year

Refuge Mean, E[exp(Xβ)] Var(exp(Xβ))

Arrowwood NWR

Woodward 1.20E+05 Woodward 1.67E+07 Brander 1.04E+11 Brander 3.15E+21

Ghermandi 258.391783 Ghermandi 21.91413

Blackwater NWR

Woodward 1.00E+05 Woodward 2.15E+07 Brander 1.32E+11 Brander 8.83E+21

Ghermandi 848.180639 Ghermandi 290.9219

Okefenokee NWR

Woodward 6.16E+04 Woodward 1.17E+07 Brander 4.73E+09 Brander 1.93E+18

Ghermandi 89.7147602 Ghermandi 3.29932

Sevilleta & Bosque del

Apache NWRs

Woodward 1.18E+05 Woodward 1.65E+07 Brander 7.30E+09 Brander 3.58E+18

Ghermandi 261.248864 Ghermandi 18.64035

65

Table 12: Water Quality Monte Carlo Mean and Variance

Water quality

dollars per acre per year

Refuge Mean, E[exp(Xβ)] Var(exp(Xβ)) Arrowwood

NWR Woodward 1.31E+05 Woodward 2.65E+07

Brander 2.40E+11 Brander 1.59E+22 Ghermandi 343.298878 Ghermandi 34.22148

Blackwater

NWR Woodward 1.05E+05 Woodward 1.81E+07

Brander 2.91E+11 Brander 4.25E+22 Ghermandi 1127.29233 Ghermandi 513.977242

Okefenokee

NWR

Woodward 6.64E+04 Woodward 1.54E+07 Brander 9.37E+09 Brander 8.79E+18

Ghermandi 119.018048 Ghermandi 4.797655909

Sevilleta & Bosque del

Apache NWRs

Woodward 1.28E+05 Woodward 2.61E+07 Brander 1.54E+10 Brander 2.72E+19

Ghermandi 348.017512 Ghermandi 31.16060726

66

Table 13: Estimated WTP per Average Acre per Year and for Refuge Wetlands per Year

` median value per

average acre NWI wetland acres median value per refuge

Woodw

ard and Wui 2001

Arrowwood NWR Flood control $1,533 4,595 $7,044,000 Water quality $1,626 4,595 $7,472,000

Blackwater NWR Flood control $845 24,502 $20,700,000 Water quality $896 24,502 $21,959,000

Okefenokee NWR Flood control $435 375,778 $163,462,000 Water quality $461 375,778 $173,393,000

Sevilleta & Bosque NWRs

Flood control $1,487 5,106 $7,594,000 Water quality $1,578 5,106 $8,056,000

Brander et al. 2006

Arrowwood NWR Flood control $2.65 4,595 $12,200 Water quality $4.33 4,595 $19,900

Blackwater NWR Flood control $13.91 24,502 $341,000 Water quality $22.70 24,502 $556,000

Okefenokee NWR Flood control $4.96 375,778 $1,863,000 Water quality $8.09 375,778 $3,042,000

Sevilleta & Bosque NWRs

Flood control $6.07 5,106 $31,000 Water quality $9.91 5,106 $50,600

Gherm

andi et al. 2010

Arrowwood NWR Flood control $17.48 4,595 $80,300 Water quality $23.48 4,595 $107,900

Blackwater NWR Flood control $46.16 24,502 $1,131,000 Water quality $62.00 24,502 $1,519,000

Okefenokee NWR Flood control $5.27 375,778 $1,979,000 Water quality $7.07 375,778 $2,659,000

Sevilleta & Bosque NWRs

Flood control $18.44 5,106 $94,100 Water quality $24.76 5,106 $126,400

67

Table 14: OLS MA Regression Model Results Variable OLS Coefficient standard error T-statistic P-value Intercept 24.147 19.863 1.216 0.23

Study/Site Acres valued† -0.583* 0.306* -1.904 0.06 Population† -0.41*** 0.148*** -2.761 0.01

Methodological (1/0) Valuation Approach

Revealed preference -4.401* 2.319* -1.898 0.06 Stated preference 0.864 1.305 0.662 0.51

Joint valuation (1,2,3) -5.977*** 1.435*** -4.164 0 Service Valued

Water quality 5.969** 2.864** 2.084 0.04 Flood protection 5.438* 2.861* 1.9 0.06

Total value 6.534** 2.836** 2.304 0.02 Recreation, general 10.667*** 2.627*** 4.061 0

Habitat 6.564** 3.093** 2.122 0.04 Recreation, fishing 5.287** 2.178** 2.428 0.02 Recreation, hunting 5.011* 2.628* 1.907 0.06

Interaction_use -1.211** 0.562** -2.153 0.04 Interaction_passive -0.705 0.56 -1.258 0.21

Context

GDP(state)† -2.681 2.076 -1.291 0.2 Coastal (1/0) 1.214** 0.566** 2.144 0.04

GDP*local_pop† 0.038 0.029 1.309 0.2 Local_pop:local_wet† 2.394** 1.124** 2.13 0.04

R2 = 0.81 Significance levels: ***<0.01, **<0.05, *<0.1

68

Table 15: PLWLS Estimated Correspondence Parameters

Correspondence Attribute Estimated Correspondence Parameter

Flood control 1.365992 GDP 1.276706

Water quality 0.760224 Coastal 0.218638

Distance 103km 0.194442 Population 0.133756

Acres 0

69

Table 16: Arrowwood NWR Stated Preference Water Quality, Ten Highest Correspondence Observations

Arrowwood NWR Stated Pref. Water quality $42,500 0 km 3,531,251 N/A

Author-date Method Service valued

GDP per capita

Euclidean distance

Study pop.

Regression Weight

Bishop et al. 2000 Stated pref. Water quality $36,200 917 936,090 2.7697

Bishop et al. 2000 Stated pref. Water quality $36,200 917 936,090 2.7697

Bishop et al. 2000 Stated pref. Water quality $36,200 917 936,090 2.7697

Sutherland & Walsh 1985

Stated pref.

Water quality, use

value $27,000 1162 1,275,514 1.8876

Sutherland & Walsh 1985

Stated pref.

Water quality bequest value $27,000 1162 1,275,514 1.8876

Sutherland & Walsh 1985

Stated pref.

Water quality exist. value $27,000 1162 1,275,514 1.8876

Sutherland & Walsh 1985

Stated pref.

Water quality option value $27,000 1162 1,275,514 1.8876

Sutherland & Walsh 1985

Stated pref.

Water quality total value $27,000 1162 1,275,514 1.8876

Phaneuf & Herriges 1999

Revealed pref. Recreation $32,300 645 630,561 1.3689

Phaneuf & Herriges 1999

Revealed pref. Recreation $32,300 721 630,561 1.3488

70

Table 17: Blackwater NWR Stated Preference Water Quality, Ten Highest Correspondence Observations

Blackwater NWR Stated Pref. Water quality $44,000 0 km 3,531,251 N/A

Author-date Method Service valued

GDP per capita

Euclidean distance

Study pop.

Regression Weight

Bishop et al. 2000 Stated pref. Water quality $36,200 1180 936,090 2.8578

Bishop et al. 2000 Stated pref. Water quality $36,200 1180 936,090 2.8578

Bishop et al. 2000 Stated pref. Water quality $36,200 1180 936,090 2.8578

Beran 1995 Stated pref. Total value $43,300 695 1,258,753 1.9196

Petrolia & Kim 2011

Stated pref. Total value $36,800 1702 2,146,273 1.7119

Bauer, Cyr, & Swallow 2004

Stated pref. Total value $34,800 512 408,000 1.6123

Bauer, Cyr, & Swallow 2004

Stated pref. Total value $34,800 512 408,000 1.6123

Bauer, Cyr, & Swallow 2004

Stated pref. Total value $34,800 512 408,000 1.6123

Bauer, Cyr, & Swallow 2004

Stated pref. Total value $34,800 512 408,000 1.6123

Johnston et al. 2002 Stated pref. Total value $38,800 433 73,423 1.4913

71

Table 18: Okefenokee NWR Stated Preference Water Quality, Ten Highest Correspondence Observations

Okefenokee NWR Stated Pref. Water quality $36,400 0 km 3,531,251 N/A

Author-date Method Service valued

GDP per capita

Euclidean distance

Study pop.

Regression Weight

Bishop et al. 2000 Stated pref. Water quality $36,200 1632 936,090 2.8787

Bishop et al. 2000 Stated pref. Water quality $36,200 1632 936,090 2.8787

Bishop et al. 2000 Stated pref. Water quality $36,200 1632 936,090 2.8787

MacDonald, Bergstrom, & Houston 1998

Stated pref. Habitat $29,700 319 2,793,672 1.5584

Sutherland & Walsh 1985

Stated pref.

Water quality, use

value $27,000 3298 1,275,514 1.4882

Sutherland & Walsh 1985

Stated pref.

Water quality bequest value $27,000 3298 1,275,514 1.4882

Sutherland & Walsh 1985

Stated pref.

Water quality exist. value $27,000 3298 1,275,514 1.4882

Sutherland & Walsh 1985

Stated pref.

Water quality option value $27,000 3298 1,275,514 1.4882

Sutherland & Walsh 1985

Stated pref.

Water quality total value $27,000 3298 1,275,514 1.4882

Beran 1995 Stated pref. Total value $43,300 326 1,258,753 1.4581

72

Table 19: Sevilleta & Bosque del Apache NWRs Stated Preference Water Quality, Ten Highest

Correspondence Observations

Sevilleta & Bosque del Apache NWRs

Stated Pref. Water quality $32,900 0 km 3,531,251 N/A

Author-date Method Service valued

GDP per capita

Euclidean distance

Study pop.

Regression Weight

Bishop et al. 2000 Stated pref. Water quality $36,200 2040 936,090 1.936

Bishop et al. 2000 Stated pref. Water quality $36,200 2040 936,090 1.936

Bishop et al. 2000 Stated pref. Water quality $36,200 2040 936,090 1.936

Sutherland & Walsh 1985

Stated pref.

Water quality, use

value $27,000 1662 1,275,514 1.8982

Sutherland & Walsh 1985

Stated pref.

Water quality bequest value $27,000 1662 1,275,514 1.8982

Sutherland & Walsh 1985

Stated pref.

Water quality exist. value $27,000 1662 1,275,514 1.8982

Sutherland & Walsh 1985

Stated pref.

Water quality option value $27,000 1662 1,275,514 1.8982

Sutherland & Walsh 1985

Stated pref.

Water quality total value $27,000 1662 1,275,514 1.8982

Sanders, Walsh, & McKean 1991

Revealed pref. Recreation $29,600 566 2,889,964 1.3836

Sanders, Walsh, & Loomis 1990

Stated pref.

Total use value $29,600 566 2,889,964 1.3836

73

Table 20: Arrowwood NWR Stated Preference Flood control, Ten Highest Correspondence Observations

Arrowwood NWR Stated Pref. Flood control $42,500 0 km 3,531,251 N/A

Author-date Method Service valued

GDP per capita

Euclidean distance

Study pop.

Regression Weight

Leitch and Hoyde 1996

Damage avoidance Flood control $26,700 98 638,800 3.8413

Leitch and Hoyde 1996

Damage avoidance

Flood control & habitat $26,700 145 638,800 3.8062

Roberts and Leitch 1997

Damage avoidance

Flood control & habitat $31,300 233 106,406 3.6005

Roberts and Leitch 1997

Damage avoidance Flood control $31,300 235 106,406 3.5993

Costanza, Farber, & Maxwell 1989

Damage avoidance Flood control $23,800 2116 94,393 1.3931

Phaneuf & Herriges 1999

Revealed pref. Recreation $32,300 645 630,561 1.371

Phaneuf & Herriges 1999

Revealed pref. Recreation $32,300 721 630,561 1.3509

Poor 1999 Stated pref. Total value $33,200 739 1,541,253 1.2821

Beran 1995 Stated pref. Total value $43,300 2205 1,258,753 1.2542

Sanders, Walsh, & Loomis 1990

Stated pref.

Total use value $29,600 1034 2,889,964 1.1417

74

Table 21: Blackwater NWR Stated Preference Flood control, Ten Highest Correspondence Observations

Blackwater NWR Stated Pref. Flood control $44,000 0 km 3,531,251 N/A

Author-date Method Service valued

GDP per capita

Euclidean distance

Study pop.

Regression Weight

Leitch and Hoyde 1996

Damage avoidance Flood control $26,700 1991 638,800 2.9085

Roberts and Leitch 1997

Damage avoidance

Flood control & habitat $31,300 1867 106,406 2.8671

Roberts and Leitch 1997

Damage avoidance Flood control $31,300 1872 106,406 2.864

Leitch and Hoyde 1996

Damage avoidance

Flood control & habitat $26,700 2191 638,800 2.7972

Costanza, Farber, & Maxwell 1989

Damage avoidance Flood control $23,800 1689 94,393 2.5647

Beran 1995 Stated pref. Total value $43,300 695 1,258,753 1.937

Petrolia & Kim 2011

Stated pref. Total value $36,800 1702 2,146,273 1.7274

Bauer, Cyr, & Swallow 2004

Stated pref. Total value $34,800 512 408,000 1.627

Bauer, Cyr, & Swallow 2004

Stated pref. Total value $34,800 512 408,000 1.627

Bauer, Cyr, & Swallow 2004

Stated pref. Total value $34,800 512 408,000 1.627

75

Table 22: Okefenokee NWR Stated Preference Flood control, Ten Highest Correspondence Observations

Okefenokee NWR Stated Pref. Flood control $36,400 0 km 3,531,251 N/A

Author-date Method Service valued

GDP per capita

Euclidean distance

Study pop.

Regression Weight

Leitch and Hoyde 1996

Damage avoidance Flood control $26,700 2217 638,800 3.0917

Roberts and Leitch 1997

Damage avoidance Flood control $31,300 2079 106,406 3.0558

Roberts and Leitch 1997

Damage avoidance

Flood control & habitat $31,300 2082 106,406 3.0541

Leitch and Hoyde 1996

Damage avoidance

Flood control & habitat $26,700 2453 638,800 2.953

Costanza, Farber, & Maxwell 1989

Damage avoidance Flood control $23,800 839 94,393 2.1704

MacDonald, Bergstrom, & Houston 1998

Stated pref. Habitat $29,700 319 2,793,672 1.5883

Beran 1995 Stated pref. Total value $43,300 326 1,258,753 1.4861

Petrolia & Kim 2011

Stated pref. Total value $36,800 870 2,146,273 1.422

Phaneuf & Herriges 1999

Revealed pref. Recreation $32,300 1596 630,561 1.3848

Phaneuf & Herriges 1999

Revealed pref. Recreation $32,300 1674 630,561 1.3639

76

Table 23: Sevilleta & Bosque del Apache NWRs Stated Preference Flood control, Ten Highest

Correspondence Observations

Sevilleta & Bosque del Apache NWRs

Stated Pref. Flood control $32,900 0 km 3,531,251 N/A

Author-date Method Service valued

GDP per capita

Euclidean distance

Study pop.

Regression Weight

Leitch and Hoyde 1996

Damage avoidance Flood control $26,700 1604 638,800 3.1907

Leitch and Hoyde 1996

Damage avoidance

Flood control & habitat $26,700 1714 638,800 3.123

Roberts and Leitch 1997

Damage avoidance Flood control $31,300 1563 106,406 3.0947

Roberts and Leitch 1997

Damage avoidance

Flood control & habitat $31,300 1580 106,406 3.0848

Costanza, Farber, & Maxwell 1989

Damage avoidance Flood control $23,800 1600 94,393 1.7147

Sanders, Walsh, & McKean 1991

Revealed pref. Recreation $29,600 566 2,889,964 1.3919

Sanders, Walsh, & Loomis 1990

Stated pref.

Total use value $29,600 566 2,889,964 1.3919

Sanders, Walsh, & Loomis 1990

Stated pref.

Total exist. value $29,600 566 2,889,964 1.3919

Sanders, Walsh, & Loomis 1990

Stated pref.

Total option value $29,600 566 2,889,964 1.3919

Sanders, Walsh, & Loomis 1990

Stated pref.

Total bequest value $29,600 566 2,889,964 1.3919

77

Table 24: OLS and PLWLS Forecasts for 4 NWRs, Annual dollars per 1000 Acres per 1000 People per

Year

2010 US dollars per 1000 acres per 1000

people per year OLS PLWLS

Site Service Valued

Median Mean Median Mean exp(XBi) exp(XBi) exp(XBi) exp(XBi)

Arrowwood NWR

WQ $170 $520 $370 $450 FC $100 $310 $110 $130

Blackwater NWR

WQ $720 $2,210 $290 $330 FC $420 $1,300 $490 $550

Okefenokee NWR

WQ $160 $480 $80 $90 FC $90 $280 $180 $210

Sevilleta & Bosque del

Apache NWRs

WQ $320 $970 $810 $980

FC $190 $570 $210 $260

78

Table 25: OLS and PLWLS Aggregate Values for 4 NWRs

Aggregate Annual Value 2010 US Dollars per Refuge per Year

Site Service Valued acres

OLS OLS PLWLS median

PLWLS mean median mean

Arrowwood NWR

WQ 4,595 2,767,000 8,508,000 6,043,000 7,229,000

FC 1,626,000 5,000,000 1,830,000 2,165,000

Blackwater NWR

WQ 24,502 62,294,000 191,523,000 25,479,000 28,748,000

FC 36,606,000 112,547,000 42,653,000 47,530,000

Okefenokee NWR

WQ 375,778 208,374,000 640,647,000 101,817,000 120,588,000 FC 122,450,000 376,474,000 238,576,000 279,353,000

Sevilleta & Bosque del

Apache NWRs

WQ 5,106 5,682,000 17,468,000 14,551,000 17,661,000

FC 3,339,000 10,265,000 3,838,000 4,625,000

79

Table 26: Comparison of 5 MA Models for 4 NWRs, Flood Control 2010 US Dollars per Year for NWR Wetlands

Arrowwood NWR Flood control

Woodward Brander Ghermandi OLS PLWLS Median 7,043,776 12,177 80,325 1,626,115 1,830,089 Mean 553,429,493 4.780E+14 1,187,381 4,999,561 2,165,016

Blackwater NWR Flood control

Woodward Brander Ghermandi OLS PLWLS Median 20,700,411 340,817 1,130,993 36,606,278 42,653,249 Mean 2,450,105,399 3.228E+15 20,781,772 112,546,849 47,529,586

Okefenokee NWR Flood control

Woodward Brander Ghermandi OLS PLWLS Median 163,463,577 1,863,861 1,980,352 122,449,919 238,575,511 Mean 2.32E+10 1.78E+15 33,712,864 376,473,987 279,353,226

Sevilleta & Bosque del Apache NWRs Flood control

Woodward Brander Ghermandi OLS PLWLS Median 7,594,007 30,992 94,150 3,338,566 3,837,809 Mean 603,359,646 3.73E+13 1,333,875 10,264,508 4,624,627

80

Table 27 Comparison of 5 MA Models for 4 NWRs, Water Quality

2010 US Dollars per Year for NWR Wetlands Arrowwood NWR

Water quality Woodward Brander Ghermandi OLS PLWLS

Median 7,471,872 19,898 107,897 2,767,365 6,043,124 Mean 600,924,752 1.102E+15 1,577,553 8,508,016 7,229,323

Blackwater NWR Water quality

Woodward Brander Ghermandi OLS PLWLS Median 21,958,813 556,186 1,519,098 62,294,439 25,478,779 Mean 2,581,881,606 7.131E+15 27,620,452 191,523,451 28,747,552

Okefenokee NWR Water quality

Woodward Brander Ghermandi OLS PLWLS Median 173,391,641 3,040,047 2,656,753 208,373,728 101,816,900 Mean 2.50E+10 3.52E+15 44,724,404 640,646,704 120,588,184

Sevilleta & Bosque del Apache NWRs Water quality

Woodward Brander Ghermandi OLS PLWLS Median 8,055,619 50,598 126,419 5,681,530 14,550,536 Mean 655,118,693 7.85E+13 1,776,895 17,467,567 17,661,387